diff --git a/folding-schemes/src/folding/hypernova/cccs.rs b/folding-schemes/src/folding/hypernova/cccs.rs
index 5b95df2..0a253a8 100644
--- a/folding-schemes/src/folding/hypernova/cccs.rs
+++ b/folding-schemes/src/folding/hypernova/cccs.rs
@@ -9,9 +9,11 @@ use std::sync::Arc;
 
 use ark_std::rand::Rng;
 
+use super::circuits::CCCSVar;
 use super::Witness;
 use crate::arith::{ccs::CCS, Arith};
 use crate::commitment::CommitmentScheme;
+use crate::folding::traits::CommittedInstanceOps;
 use crate::transcript::AbsorbNonNative;
 use crate::utils::mle::dense_vec_to_dense_mle;
 use crate::utils::vec::mat_vec_mul;
@@ -126,9 +128,8 @@ impl<C: CurveGroup> Absorb for CCCS<C>
 where
     C::ScalarField: Absorb,
 {
-    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
-        // This is never called
-        unimplemented!()
+    fn to_sponge_bytes(&self, dest: &mut Vec<u8>) {
+        C::ScalarField::batch_to_sponge_bytes(&self.to_sponge_field_elements_as_vec(), dest);
     }
 
     fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
@@ -142,6 +143,18 @@ where
     }
 }
 
+impl<C: CurveGroup> CommittedInstanceOps<C> for CCCS<C> {
+    type Var = CCCSVar<C>;
+
+    fn get_commitments(&self) -> Vec<C> {
+        vec![self.C]
+    }
+
+    fn is_incoming(&self) -> bool {
+        true
+    }
+}
+
 #[cfg(test)]
 pub mod tests {
     use ark_pallas::Fr;
diff --git a/folding-schemes/src/folding/hypernova/circuits.rs b/folding-schemes/src/folding/hypernova/circuits.rs
index 1c2d786..1fb4f09 100644
--- a/folding-schemes/src/folding/hypernova/circuits.rs
+++ b/folding-schemes/src/folding/hypernova/circuits.rs
@@ -1,6 +1,6 @@
 /// Implementation of [HyperNova](https://eprint.iacr.org/2023/573.pdf) circuits
 use ark_crypto_primitives::sponge::{
-    constraints::CryptographicSpongeVar,
+    constraints::{AbsorbGadget, CryptographicSpongeVar},
     poseidon::{constraints::PoseidonSpongeVar, PoseidonSponge},
     CryptographicSponge,
 };
@@ -14,6 +14,7 @@ use ark_r1cs_std::{
     fields::{fp::FpVar, FieldVar},
     groups::GroupOpsBounds,
     prelude::CurveVar,
+    uint8::UInt8,
     R1CSVar, ToConstraintFieldGadget,
 };
 use ark_relations::r1cs::{
@@ -41,6 +42,7 @@ use crate::folding::{
         CF1, CF2,
     },
     nova::get_r1cs_from_cs,
+    traits::CommittedInstanceVarOps,
 };
 use crate::frontend::FCircuit;
 use crate::utils::virtual_polynomial::VPAuxInfo;
@@ -52,19 +54,16 @@ use crate::{
 
 /// Committed CCS instance
 #[derive(Debug, Clone)]
-pub struct CCCSVar<C: CurveGroup>
-where
-    <C as CurveGroup>::BaseField: PrimeField,
-{
+pub struct CCCSVar<C: CurveGroup> {
     // Commitment to witness
     pub C: NonNativeAffineVar<C>,
     // Public io
     pub x: Vec<FpVar<CF1<C>>>,
 }
+
 impl<C> AllocVar<CCCS<C>, CF1<C>> for CCCSVar<C>
 where
     C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: PrimeField,
 {
     fn new_variable<T: Borrow<CCCS<C>>>(
         cs: impl Into<Namespace<CF1<C>>>,
@@ -83,12 +82,30 @@ where
     }
 }
 
+impl<C: CurveGroup> CommittedInstanceVarOps<C> for CCCSVar<C> {
+    type PointVar = NonNativeAffineVar<C>;
+
+    fn get_commitments(&self) -> Vec<Self::PointVar> {
+        vec![self.C.clone()]
+    }
+
+    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>] {
+        &self.x
+    }
+
+    fn enforce_incoming(&self) -> Result<(), SynthesisError> {
+        // `CCCSVar` is always the incoming instance
+        Ok(())
+    }
+
+    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.x.enforce_equal(&other.x)
+    }
+}
+
 /// Linearized Committed CCS instance
 #[derive(Debug, Clone)]
-pub struct LCCCSVar<C: CurveGroup>
-where
-    <C as CurveGroup>::BaseField: PrimeField,
-{
+pub struct LCCCSVar<C: CurveGroup> {
     // Commitment to witness
     pub C: NonNativeAffineVar<C>,
     // Relaxation factor of z for folded LCCCS
@@ -100,10 +117,10 @@ where
     // Vector of v_i
     pub v: Vec<FpVar<CF1<C>>>,
 }
+
 impl<C> AllocVar<LCCCS<C>, CF1<C>> for LCCCSVar<C>
 where
     C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: PrimeField,
 {
     fn new_variable<T: Borrow<LCCCS<C>>>(
         cs: impl Into<Namespace<CF1<C>>>,
@@ -127,41 +144,44 @@ where
     }
 }
 
-impl<C> LCCCSVar<C>
-where
-    C: CurveGroup,
-    <C as Group>::ScalarField: Absorb,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-{
-    /// [`LCCCSVar`].hash implements the LCCCS instance hash compatible with the native
-    /// implementation from LCCCS.hash.
-    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U` is the LCCCS.
-    /// Additionally it returns the vector of the field elements from the self parameters, so they
-    /// can be reused in other gadgets avoiding recalculating (reconstraining) them.
-    #[allow(clippy::type_complexity)]
-    pub fn hash(
-        self,
-        sponge: &PoseidonSpongeVar<CF1<C>>,
-        pp_hash: FpVar<CF1<C>>,
-        i: FpVar<CF1<C>>,
-        z_0: Vec<FpVar<CF1<C>>>,
-        z_i: Vec<FpVar<CF1<C>>>,
-    ) -> Result<(FpVar<CF1<C>>, Vec<FpVar<CF1<C>>>), SynthesisError> {
-        let mut sponge = sponge.clone();
-        let U_vec = [
-            self.C.to_constraint_field()?,
-            vec![self.u],
-            self.x,
-            self.r_x,
-            self.v,
+impl<C: CurveGroup> AbsorbGadget<C::ScalarField> for LCCCSVar<C> {
+    fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
+        FpVar::batch_to_sponge_bytes(&self.to_sponge_field_elements()?)
+    }
+
+    fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
+        Ok([
+            &self.C.to_constraint_field()?,
+            &[self.u.clone()][..],
+            &self.x,
+            &self.r_x,
+            &self.v,
         ]
-        .concat();
-        sponge.absorb(&pp_hash)?;
-        sponge.absorb(&i)?;
-        sponge.absorb(&z_0)?;
-        sponge.absorb(&z_i)?;
-        sponge.absorb(&U_vec)?;
-        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
+        .concat())
+    }
+}
+
+impl<C: CurveGroup> CommittedInstanceVarOps<C> for LCCCSVar<C> {
+    type PointVar = NonNativeAffineVar<C>;
+
+    fn get_commitments(&self) -> Vec<Self::PointVar> {
+        vec![self.C.clone()]
+    }
+
+    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>] {
+        &self.x
+    }
+
+    fn enforce_incoming(&self) -> Result<(), SynthesisError> {
+        // `LCCCSVar` is always the running instance
+        Err(SynthesisError::Unsatisfiable)
+    }
+
+    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.u.enforce_equal(&other.u)?;
+        self.x.enforce_equal(&other.x)?;
+        self.r_x.enforce_equal(&other.r_x)?;
+        self.v.enforce_equal(&other.v)
     }
 }
 
@@ -742,25 +762,13 @@ where
 
         let sponge = PoseidonSpongeVar::<C1::ScalarField>::new(cs.clone(), &self.poseidon_config);
 
-        // get z_{i+1} from the F circuit
-        let i_usize = self.i_usize.unwrap_or(0);
-        let z_i1 =
-            self.F
-                .generate_step_constraints(cs.clone(), i_usize, z_i.clone(), external_inputs)?;
-
         let is_basecase = i.is_zero()?;
         let is_not_basecase = is_basecase.not();
 
         // Primary Part
         // P.1. Compute u_i.x
         // u_i.x[0] = H(i, z_0, z_i, U_i)
-        let (u_i_x, _) = U_i.clone().hash(
-            &sponge,
-            pp_hash.clone(),
-            i.clone(),
-            z_0.clone(),
-            z_i.clone(),
-        )?;
+        let (u_i_x, _) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
         // u_i.x[1] = H(cf_U_i)
         let (cf_u_i_x, cf_U_i_vec) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
 
@@ -795,19 +803,26 @@ where
         U_i1.C = U_i1_C;
 
         // P.4.a compute and check the first output of F'
+
+        // get z_{i+1} from the F circuit
+        let i_usize = self.i_usize.unwrap_or(0);
+        let z_i1 = self
+            .F
+            .generate_step_constraints(cs.clone(), i_usize, z_i, external_inputs)?;
+
         let (u_i1_x, _) = U_i1.clone().hash(
             &sponge,
-            pp_hash.clone(),
-            i + FpVar::<CF1<C1>>::one(),
-            z_0.clone(),
-            z_i1.clone(),
+            &pp_hash,
+            &(i + FpVar::<CF1<C1>>::one()),
+            &z_0,
+            &z_i1,
         )?;
         let (u_i1_x_base, _) = LCCCSVar::new_constant(cs.clone(), U_dummy)?.hash(
             &sponge,
-            pp_hash.clone(),
-            FpVar::<CF1<C1>>::one(),
-            z_0.clone(),
-            z_i1.clone(),
+            &pp_hash,
+            &FpVar::<CF1<C1>>::one(),
+            &z_0,
+            &z_i1,
         )?;
         let x = FpVar::new_input(cs.clone(), || Ok(self.x.unwrap_or(u_i1_x_base.value()?)))?;
         x.enforce_equal(&is_basecase.select(&u_i1_x_base, &u_i1_x)?)?;
@@ -900,6 +915,7 @@ mod tests {
                 utils::{compute_c, compute_sigmas_thetas},
                 HyperNovaCycleFoldCircuit,
             },
+            traits::CommittedInstanceOps,
         },
         frontend::utils::CubicFCircuit,
         transcript::poseidon::poseidon_canonical_config,
@@ -1113,6 +1129,37 @@ mod tests {
         assert_eq!(folded_lcccsVar.u.value().unwrap(), folded_lcccs.u);
     }
 
+    /// test that checks the native LCCCS.to_sponge_{bytes,field_elements} vs
+    /// the R1CS constraints version
+    #[test]
+    pub fn test_lcccs_to_sponge_preimage() {
+        let mut rng = test_rng();
+
+        let ccs = get_test_ccs();
+        let z1 = get_test_z::<Fr>(3);
+
+        let (pedersen_params, _) =
+            Pedersen::<Projective>::setup(&mut rng, ccs.n - ccs.l - 1).unwrap();
+
+        let (lcccs, _) = ccs
+            .to_lcccs::<_, _, Pedersen<Projective, true>, true>(&mut rng, &pedersen_params, &z1)
+            .unwrap();
+        let bytes = lcccs.to_sponge_bytes_as_vec();
+        let field_elements = lcccs.to_sponge_field_elements_as_vec();
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        let lcccsVar = LCCCSVar::<Projective>::new_witness(cs.clone(), || Ok(lcccs)).unwrap();
+        let bytes_var = lcccsVar.to_sponge_bytes().unwrap();
+        let field_elements_var = lcccsVar.to_sponge_field_elements().unwrap();
+
+        assert!(cs.is_satisfied().unwrap());
+
+        // check that the natively computed and in-circuit computed hashes match
+        assert_eq!(bytes_var.value().unwrap(), bytes);
+        assert_eq!(field_elements_var.value().unwrap(), field_elements);
+    }
+
     /// test that checks the native LCCCS.hash vs the R1CS constraints version
     #[test]
     pub fn test_lcccs_hash() {
@@ -1133,9 +1180,7 @@ mod tests {
         let (lcccs, _) = ccs
             .to_lcccs::<_, _, Pedersen<Projective, true>, true>(&mut rng, &pedersen_params, &z1)
             .unwrap();
-        let h = lcccs
-            .clone()
-            .hash(&sponge, pp_hash, i, z_0.clone(), z_i.clone());
+        let h = lcccs.clone().hash(&sponge, pp_hash, i, &z_0, &z_i);
 
         let cs = ConstraintSystem::<Fr>::new_ref();
 
@@ -1147,13 +1192,7 @@ mod tests {
         let lcccsVar = LCCCSVar::<Projective>::new_witness(cs.clone(), || Ok(lcccs)).unwrap();
         let (hVar, _) = lcccsVar
             .clone()
-            .hash(
-                &spongeVar,
-                pp_hashVar,
-                iVar.clone(),
-                z_0Var.clone(),
-                z_iVar.clone(),
-            )
+            .hash(&spongeVar, &pp_hashVar, &iVar, &z_0Var, &z_iVar)
             .unwrap();
         assert!(cs.is_satisfied().unwrap());
 
@@ -1225,7 +1264,7 @@ mod tests {
         let mut cf_W_i = cf_W_dummy.clone();
         let mut cf_U_i = cf_U_dummy.clone();
         u_i.x = vec![
-            U_i.hash(&sponge, pp_hash, Fr::zero(), z_0.clone(), z_i.clone()),
+            U_i.hash(&sponge, pp_hash, Fr::zero(), &z_0, &z_i),
             cf_U_i.hash_cyclefold(&sponge, pp_hash),
         ];
 
@@ -1252,7 +1291,7 @@ mod tests {
                 W_i1 = Witness::<Fr>::dummy(&ccs);
                 U_i1 = LCCCS::dummy(ccs.l, ccs.t, ccs.s);
 
-                let u_i1_x = U_i1.hash(&sponge, pp_hash, Fr::one(), z_0.clone(), z_i1.clone());
+                let u_i1_x = U_i1.hash(&sponge, pp_hash, Fr::one(), &z_0, &z_i1);
 
                 // hash the initial (dummy) CycleFold instance, which is used as the 2nd public
                 // input in the AugmentedFCircuit
@@ -1309,8 +1348,7 @@ mod tests {
                 // sanity check: check the folded instance relation
                 U_i1.check_relation(&ccs, &W_i1).unwrap();
 
-                let u_i1_x =
-                    U_i1.hash(&sponge, pp_hash, iFr + Fr::one(), z_0.clone(), z_i1.clone());
+                let u_i1_x = U_i1.hash(&sponge, pp_hash, iFr + Fr::one(), &z_0, &z_i1);
 
                 let rho_bits = rho.into_bigint().to_bits_le()[..NOVA_N_BITS_RO].to_vec();
                 let rho_Fq = Fq::from_bigint(BigInteger::from_bits_le(&rho_bits)).unwrap();
@@ -1434,8 +1472,7 @@ mod tests {
             assert_eq!(u_i.x, r1cs_x_i1);
             assert_eq!(u_i.x[0], augmented_f_circuit.x.unwrap());
             assert_eq!(u_i.x[1], augmented_f_circuit.cf_x.unwrap());
-            let expected_u_i1_x =
-                U_i1.hash(&sponge, pp_hash, iFr + Fr::one(), z_0.clone(), z_i1.clone());
+            let expected_u_i1_x = U_i1.hash(&sponge, pp_hash, iFr + Fr::one(), &z_0, &z_i1);
             let expected_cf_U_i1_x = cf_U_i.hash_cyclefold(&sponge, pp_hash);
             // u_i is already u_i1 at this point, check that has the expected value at x[0]
             assert_eq!(u_i.x[0], expected_u_i1_x);
diff --git a/folding-schemes/src/folding/hypernova/decider_eth_circuit.rs b/folding-schemes/src/folding/hypernova/decider_eth_circuit.rs
index b4e2cf8..5f367fb 100644
--- a/folding-schemes/src/folding/hypernova/decider_eth_circuit.rs
+++ b/folding-schemes/src/folding/hypernova/decider_eth_circuit.rs
@@ -26,8 +26,6 @@ use super::{
     nimfs::{NIMFSProof, NIMFS},
     HyperNova, Witness, CCCS, LCCCS,
 };
-use crate::arith::ccs::CCS;
-use crate::arith::r1cs::R1CS;
 use crate::commitment::{pedersen::Params as PedersenParams, CommitmentScheme};
 use crate::folding::circuits::{
     cyclefold::{CycleFoldCommittedInstance, CycleFoldWitness},
@@ -40,6 +38,8 @@ use crate::utils::{
     vec::poly_from_vec,
 };
 use crate::Error;
+use crate::{arith::ccs::CCS, folding::traits::CommittedInstanceVarOps};
+use crate::{arith::r1cs::R1CS, folding::traits::WitnessVarOps};
 
 /// In-circuit representation of the Witness associated to the CommittedInstance.
 #[derive(Debug, Clone)]
@@ -66,6 +66,12 @@ impl<F: PrimeField> AllocVar<Witness<F>, F> for WitnessVar<F> {
     }
 }
 
+impl<F: PrimeField> WitnessVarOps<F> for WitnessVar<F> {
+    fn get_openings(&self) -> Vec<(&[FpVar<F>], FpVar<F>)> {
+        vec![(&self.w, self.r_w.clone())]
+    }
+}
+
 /// CCSMatricesVar contains the matrices 'M' of the CCS without the rest of CCS parameters.
 #[derive(Debug, Clone)]
 pub struct CCSMatricesVar<F: PrimeField> {
@@ -340,13 +346,7 @@ where
         )?;
 
         // 3.a u_i.x[0] == H(i, z_0, z_i, U_i)
-        let (u_i_x, _) = U_i.clone().hash(
-            &sponge,
-            pp_hash.clone(),
-            i.clone(),
-            z_0.clone(),
-            z_i.clone(),
-        )?;
+        let (u_i_x, _) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
         (u_i.x[0]).enforce_equal(&u_i_x)?;
 
         #[cfg(feature = "light-test")]
diff --git a/folding-schemes/src/folding/hypernova/lcccs.rs b/folding-schemes/src/folding/hypernova/lcccs.rs
index 6aa1652..e95fd8c 100644
--- a/folding-schemes/src/folding/hypernova/lcccs.rs
+++ b/folding-schemes/src/folding/hypernova/lcccs.rs
@@ -1,5 +1,5 @@
 use ark_crypto_primitives::sponge::Absorb;
-use ark_ec::{CurveGroup, Group};
+use ark_ec::CurveGroup;
 use ark_ff::PrimeField;
 use ark_poly::DenseMultilinearExtension;
 use ark_poly::MultilinearExtension;
@@ -8,10 +8,12 @@ use ark_serialize::CanonicalSerialize;
 use ark_std::rand::Rng;
 use ark_std::Zero;
 
+use super::circuits::LCCCSVar;
 use super::Witness;
 use crate::arith::ccs::CCS;
 use crate::commitment::CommitmentScheme;
-use crate::transcript::{AbsorbNonNative, Transcript};
+use crate::folding::traits::CommittedInstanceOps;
+use crate::transcript::AbsorbNonNative;
 use crate::utils::mle::dense_vec_to_dense_mle;
 use crate::utils::vec::mat_vec_mul;
 use crate::Error;
@@ -121,9 +123,8 @@ impl<C: CurveGroup> Absorb for LCCCS<C>
 where
     C::ScalarField: Absorb,
 {
-    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
-        // This is never called
-        unimplemented!()
+    fn to_sponge_bytes(&self, dest: &mut Vec<u8>) {
+        C::ScalarField::batch_to_sponge_bytes(&self.to_sponge_field_elements_as_vec(), dest);
     }
 
     fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
@@ -140,29 +141,15 @@ where
     }
 }
 
-impl<C: CurveGroup> LCCCS<C>
-where
-    <C as Group>::ScalarField: Absorb,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-{
-    /// [`LCCCS`].hash implements the committed instance hash compatible with the gadget
-    /// implemented in nova/circuits.rs::CommittedInstanceVar.hash.
-    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U_i` is the LCCCS.
-    pub fn hash<T: Transcript<C::ScalarField>>(
-        &self,
-        sponge: &T,
-        pp_hash: C::ScalarField,
-        i: C::ScalarField,
-        z_0: Vec<C::ScalarField>,
-        z_i: Vec<C::ScalarField>,
-    ) -> C::ScalarField {
-        let mut sponge = sponge.clone();
-        sponge.absorb(&pp_hash);
-        sponge.absorb(&i);
-        sponge.absorb(&z_0);
-        sponge.absorb(&z_i);
-        sponge.absorb(&self);
-        sponge.squeeze_field_elements(1)[0]
+impl<C: CurveGroup> CommittedInstanceOps<C> for LCCCS<C> {
+    type Var = LCCCSVar<C>;
+
+    fn get_commitments(&self) -> Vec<C> {
+        vec![self.C]
+    }
+
+    fn is_incoming(&self) -> bool {
+        false
     }
 }
 
diff --git a/folding-schemes/src/folding/hypernova/mod.rs b/folding-schemes/src/folding/hypernova/mod.rs
index 2917ea4..c63d943 100644
--- a/folding-schemes/src/folding/hypernova/mod.rs
+++ b/folding-schemes/src/folding/hypernova/mod.rs
@@ -20,6 +20,7 @@ pub mod utils;
 
 use cccs::CCCS;
 use circuits::AugmentedFCircuit;
+use decider_eth_circuit::WitnessVar;
 use lcccs::LCCCS;
 use nimfs::NIMFS;
 
@@ -32,6 +33,7 @@ use crate::folding::circuits::{
     CF2,
 };
 use crate::folding::nova::{get_r1cs_from_cs, PreprocessorParam};
+use crate::folding::traits::{CommittedInstanceOps, WitnessOps};
 use crate::frontend::FCircuit;
 use crate::utils::{get_cm_coordinates, pp_hash};
 use crate::Error;
@@ -81,6 +83,14 @@ impl<F: PrimeField> Witness<F> {
     }
 }
 
+impl<F: PrimeField> WitnessOps<F> for Witness<F> {
+    type Var = WitnessVar<F>;
+
+    fn get_openings(&self) -> Vec<(&[F], F)> {
+        vec![(&self.w, self.r_w)]
+    }
+}
+
 /// Proving parameters for HyperNova-based IVC
 #[derive(Debug, Clone)]
 pub struct ProverParams<C1, C2, CS1, CS2, const H: bool>
@@ -307,8 +317,8 @@ where
                 &sponge,
                 self.pp_hash,
                 C1::ScalarField::zero(), // i
-                self.z_0.clone(),
-                state.clone(),
+                &self.z_0,
+                &state,
             ),
             cf_U_i.hash_cyclefold(&sponge, self.pp_hash),
         ];
@@ -324,8 +334,8 @@ where
             &sponge,
             self.pp_hash,
             C1::ScalarField::one(), // i+1, where i=0
-            self.z_0.clone(),
-            z_i1.clone(),
+            &self.z_0,
+            &z_i1,
         );
 
         let cf_u_i1_x = cf_U_i.hash_cyclefold(&sponge, self.pp_hash);
@@ -494,13 +504,7 @@ where
         let (cf_W_dummy, cf_U_dummy): (CycleFoldWitness<C2>, CycleFoldCommittedInstance<C2>) =
             cf_r1cs.dummy_running_instance();
         u_dummy.x = vec![
-            U_dummy.hash(
-                &sponge,
-                pp_hash,
-                C1::ScalarField::zero(),
-                z_0.clone(),
-                z_0.clone(),
-            ),
+            U_dummy.hash(&sponge, pp_hash, C1::ScalarField::zero(), &z_0, &z_0),
             cf_U_dummy.hash_cyclefold(&sponge, pp_hash),
         ];
 
@@ -643,8 +647,8 @@ where
                 &sponge,
                 self.pp_hash,
                 C1::ScalarField::one(),
-                self.z_0.clone(),
-                z_i1.clone(),
+                &self.z_0,
+                &z_i1,
             );
 
             // hash the initial (dummy) CycleFold instance, which is used as the 2nd public
@@ -699,8 +703,8 @@ where
                 &sponge,
                 self.pp_hash,
                 self.i + C1::ScalarField::one(),
-                self.z_0.clone(),
-                z_i1.clone(),
+                &self.z_0,
+                &z_i1,
             );
 
             let rho_bits = rho.into_bigint().to_bits_le()[..NOVA_N_BITS_RO].to_vec();
@@ -884,7 +888,7 @@ where
 
         // check that u_i's output points to the running instance
         // u_i.X[0] == H(i, z_0, z_i, U_i)
-        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, z_0, z_i.clone());
+        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, &z_0, &z_i);
         if expected_u_i_x != u_i.x[0] {
             return Err(Error::IVCVerificationFail);
         }
diff --git a/folding-schemes/src/folding/mod.rs b/folding-schemes/src/folding/mod.rs
index 8f76a71..0f3a66f 100644
--- a/folding-schemes/src/folding/mod.rs
+++ b/folding-schemes/src/folding/mod.rs
@@ -2,3 +2,4 @@ pub mod circuits;
 pub mod hypernova;
 pub mod nova;
 pub mod protogalaxy;
+pub mod traits;
diff --git a/folding-schemes/src/folding/nova/circuits.rs b/folding-schemes/src/folding/nova/circuits.rs
index a506c8c..6e90460 100644
--- a/folding-schemes/src/folding/nova/circuits.rs
+++ b/folding-schemes/src/folding/nova/circuits.rs
@@ -21,7 +21,6 @@ use ark_std::{fmt::Debug, One, Zero};
 use core::{borrow::Borrow, marker::PhantomData};
 
 use super::{CommittedInstance, NovaCycleFoldConfig};
-use crate::constants::NOVA_N_BITS_RO;
 use crate::folding::circuits::{
     cyclefold::{
         CycleFoldChallengeGadget, CycleFoldCommittedInstance, CycleFoldCommittedInstanceVar,
@@ -32,15 +31,13 @@ use crate::folding::circuits::{
 };
 use crate::frontend::FCircuit;
 use crate::transcript::{AbsorbNonNativeGadget, Transcript, TranscriptVar};
+use crate::{constants::NOVA_N_BITS_RO, folding::traits::CommittedInstanceVarOps};
 
 /// CommittedInstanceVar contains the u, x, cmE and cmW values which are folded on the main Nova
 /// constraints field (E1::Fr, where E1 is the main curve). The peculiarity is that cmE and cmW are
 /// represented non-natively over the constraint field.
 #[derive(Debug, Clone)]
-pub struct CommittedInstanceVar<C: CurveGroup>
-where
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-{
+pub struct CommittedInstanceVar<C: CurveGroup> {
     pub u: FpVar<C::ScalarField>,
     pub x: Vec<FpVar<C::ScalarField>>,
     pub cmE: NonNativeAffineVar<C>,
@@ -50,7 +47,6 @@ where
 impl<C> AllocVar<CommittedInstance<C>, CF1<C>> for CommittedInstanceVar<C>
 where
     C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: PrimeField,
 {
     fn new_variable<T: Borrow<CommittedInstance<C>>>(
         cs: impl Into<Namespace<CF1<C>>>,
@@ -80,7 +76,7 @@ where
     <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
 {
     fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
-        unimplemented!()
+        FpVar::batch_to_sponge_bytes(&self.to_sponge_field_elements()?)
     }
 
     fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
@@ -94,35 +90,27 @@ where
     }
 }
 
-impl<C> CommittedInstanceVar<C>
-where
-    C: CurveGroup,
-    <C as Group>::ScalarField: Absorb,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-{
-    /// hash implements the committed instance hash compatible with the native implementation from
-    /// CommittedInstance.hash.
-    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U` is the
-    /// `CommittedInstance`.
-    /// Additionally it returns the vector of the field elements from the self parameters, so they
-    /// can be reused in other gadgets avoiding recalculating (reconstraining) them.
-    #[allow(clippy::type_complexity)]
-    pub fn hash<S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>>(
-        self,
-        sponge: &T,
-        pp_hash: FpVar<CF1<C>>,
-        i: FpVar<CF1<C>>,
-        z_0: Vec<FpVar<CF1<C>>>,
-        z_i: Vec<FpVar<CF1<C>>>,
-    ) -> Result<(FpVar<CF1<C>>, Vec<FpVar<CF1<C>>>), SynthesisError> {
-        let mut sponge = sponge.clone();
-        let U_vec = self.to_sponge_field_elements()?;
-        sponge.absorb(&pp_hash)?;
-        sponge.absorb(&i)?;
-        sponge.absorb(&z_0)?;
-        sponge.absorb(&z_i)?;
-        sponge.absorb(&U_vec)?;
-        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
+impl<C: CurveGroup> CommittedInstanceVarOps<C> for CommittedInstanceVar<C> {
+    type PointVar = NonNativeAffineVar<C>;
+
+    fn get_commitments(&self) -> Vec<Self::PointVar> {
+        vec![self.cmW.clone(), self.cmE.clone()]
+    }
+
+    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>] {
+        &self.x
+    }
+
+    fn enforce_incoming(&self) -> Result<(), SynthesisError> {
+        let zero = NonNativeUintVar::new_constant(ConstraintSystemRef::None, CF2::<C>::zero())?;
+        self.cmE.x.enforce_equal_unaligned(&zero)?;
+        self.cmE.y.enforce_equal_unaligned(&zero)?;
+        self.u.enforce_equal(&FpVar::one())
+    }
+
+    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.u.enforce_equal(&other.u)?;
+        self.x.enforce_equal(&other.x)
     }
 }
 
@@ -359,24 +347,12 @@ where
         // `transcript` is for challenge generation.
         let mut transcript = sponge.clone();
 
-        // get z_{i+1} from the F circuit
-        let i_usize = self.i_usize.unwrap_or(0);
-        let z_i1 =
-            self.F
-                .generate_step_constraints(cs.clone(), i_usize, z_i.clone(), external_inputs)?;
-
         let is_basecase = i.is_zero()?;
 
         // Primary Part
         // P.1. Compute u_i.x
         // u_i.x[0] = H(i, z_0, z_i, U_i)
-        let (u_i_x, U_i_vec) = U_i.clone().hash(
-            &sponge,
-            pp_hash.clone(),
-            i.clone(),
-            z_0.clone(),
-            z_i.clone(),
-        )?;
+        let (u_i_x, U_i_vec) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
         // u_i.x[1] = H(cf_U_i)
         let (cf_u_i_x, cf_U_i_vec) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
 
@@ -421,21 +397,28 @@ where
         U_i1.cmW = U_i1_cmW;
 
         // P.4.a compute and check the first output of F'
+
+        // get z_{i+1} from the F circuit
+        let i_usize = self.i_usize.unwrap_or(0);
+        let z_i1 = self
+            .F
+            .generate_step_constraints(cs.clone(), i_usize, z_i, external_inputs)?;
+
         // Base case: u_{i+1}.x[0] == H((i+1, z_0, z_{i+1}, U_{\bot})
         // Non-base case: u_{i+1}.x[0] == H((i+1, z_0, z_{i+1}, U_{i+1})
         let (u_i1_x, _) = U_i1.clone().hash(
             &sponge,
-            pp_hash.clone(),
-            i + FpVar::<CF1<C1>>::one(),
-            z_0.clone(),
-            z_i1.clone(),
+            &pp_hash,
+            &(i + FpVar::<CF1<C1>>::one()),
+            &z_0,
+            &z_i1,
         )?;
         let (u_i1_x_base, _) = CommittedInstanceVar::new_constant(cs.clone(), u_dummy)?.hash(
             &sponge,
-            pp_hash.clone(),
-            FpVar::<CF1<C1>>::one(),
-            z_0.clone(),
-            z_i1.clone(),
+            &pp_hash,
+            &FpVar::<CF1<C1>>::one(),
+            &z_0,
+            &z_i1,
         )?;
         let x = FpVar::new_input(cs.clone(), || Ok(self.x.unwrap_or(u_i1_x_base.value()?)))?;
         x.enforce_equal(&is_basecase.select(&u_i1_x_base, &u_i1_x)?)?;
@@ -538,6 +521,7 @@ pub mod tests {
     use crate::commitment::pedersen::Pedersen;
     use crate::folding::nova::nifs::tests::prepare_simple_fold_inputs;
     use crate::folding::nova::nifs::NIFS;
+    use crate::folding::traits::CommittedInstanceOps;
     use crate::transcript::poseidon::poseidon_canonical_config;
 
     #[test]
@@ -591,6 +575,36 @@ pub mod tests {
         assert!(cs.is_satisfied().unwrap());
     }
 
+    /// test that checks the native CommittedInstance.to_sponge_{bytes,field_elements}
+    /// vs the R1CS constraints version
+    #[test]
+    pub fn test_committed_instance_to_sponge_preimage() {
+        let mut rng = ark_std::test_rng();
+
+        let ci = CommittedInstance::<Projective> {
+            cmE: Projective::rand(&mut rng),
+            u: Fr::rand(&mut rng),
+            cmW: Projective::rand(&mut rng),
+            x: vec![Fr::rand(&mut rng); 1],
+        };
+
+        let bytes = ci.to_sponge_bytes_as_vec();
+        let field_elements = ci.to_sponge_field_elements_as_vec();
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        let ciVar =
+            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
+        let bytes_var = ciVar.to_sponge_bytes().unwrap();
+        let field_elements_var = ciVar.to_sponge_field_elements().unwrap();
+
+        assert!(cs.is_satisfied().unwrap());
+
+        // check that the natively computed and in-circuit computed hashes match
+        assert_eq!(bytes_var.value().unwrap(), bytes);
+        assert_eq!(field_elements_var.value().unwrap(), field_elements);
+    }
+
     #[test]
     fn test_committed_instance_hash() {
         let mut rng = ark_std::test_rng();
@@ -609,7 +623,7 @@ pub mod tests {
         };
 
         // compute the CommittedInstance hash natively
-        let h = ci.hash(&sponge, pp_hash, i, z_0.clone(), z_i.clone());
+        let h = ci.hash(&sponge, pp_hash, i, &z_0, &z_i);
 
         let cs = ConstraintSystem::<Fr>::new_ref();
 
@@ -624,7 +638,7 @@ pub mod tests {
 
         // compute the CommittedInstance hash in-circuit
         let (hVar, _) = ciVar
-            .hash(&sponge, pp_hashVar, iVar, z_0Var, z_iVar)
+            .hash(&sponge, &pp_hashVar, &iVar, &z_0Var, &z_iVar)
             .unwrap();
         assert!(cs.is_satisfied().unwrap());
 
diff --git a/folding-schemes/src/folding/nova/decider_eth_circuit.rs b/folding-schemes/src/folding/nova/decider_eth_circuit.rs
index 25564a1..27d83f1 100644
--- a/folding-schemes/src/folding/nova/decider_eth_circuit.rs
+++ b/folding-schemes/src/folding/nova/decider_eth_circuit.rs
@@ -27,8 +27,6 @@ use super::{
     nifs::NIFS,
     CommittedInstance, Nova, Witness,
 };
-use crate::arith::r1cs::R1CS;
-use crate::commitment::{pedersen::Params as PedersenParams, CommitmentScheme};
 use crate::folding::circuits::{
     cyclefold::{CycleFoldCommittedInstance, CycleFoldWitness},
     nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
@@ -41,6 +39,11 @@ use crate::utils::{
     vec::poly_from_vec,
 };
 use crate::Error;
+use crate::{arith::r1cs::R1CS, folding::traits::WitnessVarOps};
+use crate::{
+    commitment::{pedersen::Params as PedersenParams, CommitmentScheme},
+    folding::traits::CommittedInstanceVarOps,
+};
 
 #[derive(Debug, Clone)]
 pub struct RelaxedR1CSGadget {}
@@ -135,7 +138,6 @@ pub struct WitnessVar<C: CurveGroup> {
 impl<C> AllocVar<Witness<C>, CF1<C>> for WitnessVar<C>
 where
     C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: PrimeField,
 {
     fn new_variable<T: Borrow<Witness<C>>>(
         cs: impl Into<Namespace<CF1<C>>>,
@@ -160,6 +162,12 @@ where
     }
 }
 
+impl<C: CurveGroup> WitnessVarOps<C::ScalarField> for WitnessVar<C> {
+    fn get_openings(&self) -> Vec<(&[FpVar<C::ScalarField>], FpVar<C::ScalarField>)> {
+        vec![(&self.E, self.rE.clone()), (&self.W, self.rW.clone())]
+    }
+}
+
 /// Circuit that implements the in-circuit checks needed for the onchain (Ethereum's EVM)
 /// verification.
 #[derive(Clone, Debug)]
@@ -398,13 +406,7 @@ where
         (u_i.u.is_one()?).enforce_equal(&Boolean::TRUE)?;
 
         // 3.a u_i.x[0] == H(i, z_0, z_i, U_i)
-        let (u_i_x, U_i_vec) = U_i.clone().hash(
-            &sponge,
-            pp_hash.clone(),
-            i.clone(),
-            z_0.clone(),
-            z_i.clone(),
-        )?;
+        let (u_i_x, U_i_vec) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
         (u_i.x[0]).enforce_equal(&u_i_x)?;
 
         #[cfg(feature = "light-test")]
diff --git a/folding-schemes/src/folding/nova/mod.rs b/folding-schemes/src/folding/nova/mod.rs
index 02e6d3b..97150a0 100644
--- a/folding-schemes/src/folding/nova/mod.rs
+++ b/folding-schemes/src/folding/nova/mod.rs
@@ -13,6 +13,7 @@ use ark_std::fmt::Debug;
 use ark_std::rand::RngCore;
 use ark_std::{One, UniformRand, Zero};
 use core::marker::PhantomData;
+use decider_eth_circuit::WitnessVar;
 
 use crate::folding::circuits::cyclefold::{
     fold_cyclefold_circuit, CycleFoldCircuit, CycleFoldCommittedInstance, CycleFoldConfig,
@@ -38,9 +39,11 @@ pub mod nifs;
 pub mod serialize;
 pub mod traits;
 pub mod zk;
-use circuits::{AugmentedFCircuit, ChallengeGadget};
+use circuits::{AugmentedFCircuit, ChallengeGadget, CommittedInstanceVar};
 use nifs::NIFS;
 
+use super::traits::{CommittedInstanceOps, WitnessOps};
+
 /// Configuration for Nova's CycleFold circuit
 pub struct NovaCycleFoldConfig<C: CurveGroup> {
     _c: PhantomData<C>,
@@ -83,9 +86,8 @@ impl<C: CurveGroup> Absorb for CommittedInstance<C>
 where
     C::ScalarField: Absorb,
 {
-    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
-        // This is never called
-        unimplemented!()
+    fn to_sponge_bytes(&self, dest: &mut Vec<u8>) {
+        C::ScalarField::batch_to_sponge_bytes(&self.to_sponge_field_elements_as_vec(), dest);
     }
 
     fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
@@ -103,30 +105,15 @@ where
     }
 }
 
-impl<C: CurveGroup> CommittedInstance<C>
-where
-    <C as Group>::ScalarField: Absorb,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-{
-    /// hash implements the committed instance hash compatible with the gadget implemented in
-    /// nova/circuits.rs::CommittedInstanceVar.hash.
-    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U_i` is the
-    /// `CommittedInstance`.
-    pub fn hash<T: Transcript<C::ScalarField>>(
-        &self,
-        sponge: &T,
-        pp_hash: C::ScalarField, // public params hash
-        i: C::ScalarField,
-        z_0: Vec<C::ScalarField>,
-        z_i: Vec<C::ScalarField>,
-    ) -> C::ScalarField {
-        let mut sponge = sponge.clone();
-        sponge.absorb(&pp_hash);
-        sponge.absorb(&i);
-        sponge.absorb(&z_0);
-        sponge.absorb(&z_i);
-        sponge.absorb(&self);
-        sponge.squeeze_field_elements(1)[0]
+impl<C: CurveGroup> CommittedInstanceOps<C> for CommittedInstance<C> {
+    type Var = CommittedInstanceVar<C>;
+
+    fn get_commitments(&self) -> Vec<C> {
+        vec![self.cmW, self.cmE]
+    }
+
+    fn is_incoming(&self) -> bool {
+        self.cmE == C::zero() && self.u == One::one()
     }
 }
 
@@ -188,6 +175,14 @@ impl<C: CurveGroup> Witness<C> {
     }
 }
 
+impl<C: CurveGroup> WitnessOps<C::ScalarField> for Witness<C> {
+    type Var = WitnessVar<C>;
+
+    fn get_openings(&self) -> Vec<(&[C::ScalarField], C::ScalarField)> {
+        vec![(&self.W, self.rW), (&self.E, self.rE)]
+    }
+}
+
 #[derive(Debug, Clone)]
 pub struct PreprocessorParam<C1, C2, FC, CS1, CS2, const H: bool = false>
 where
@@ -696,8 +691,8 @@ where
             &sponge,
             self.pp_hash,
             self.i + C1::ScalarField::one(),
-            self.z_0.clone(),
-            z_i1.clone(),
+            &self.z_0,
+            &z_i1,
         );
         // u_{i+1}.x[1] = H(cf_U_{i+1})
         let cf_u_i1_x: C1::ScalarField;
@@ -907,7 +902,7 @@ where
 
         // check that u_i's output points to the running instance
         // u_i.X[0] == H(i, z_0, z_i, U_i)
-        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, z_0, z_i.clone());
+        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, &z_0, &z_i);
         if expected_u_i_x != u_i.x[0] {
             return Err(Error::IVCVerificationFail);
         }
diff --git a/folding-schemes/src/folding/nova/zk.rs b/folding-schemes/src/folding/nova/zk.rs
index 25a216a..8b5ff14 100644
--- a/folding-schemes/src/folding/nova/zk.rs
+++ b/folding-schemes/src/folding/nova/zk.rs
@@ -36,6 +36,7 @@ use ark_std::{One, Zero};
 
 use crate::{
     arith::r1cs::{RelaxedR1CS, R1CS},
+    folding::traits::CommittedInstanceOps,
     RngCore,
 };
 use ark_crypto_primitives::sponge::{
@@ -226,7 +227,7 @@ where
 
         // b. Check computed hashes are correct
         let mut sponge = PoseidonSponge::<C1::ScalarField>::new(poseidon_config);
-        let expected_u_i_x = proof.U_i.hash(&sponge, pp_hash, i, z_0, z_i);
+        let expected_u_i_x = proof.U_i.hash(&sponge, pp_hash, i, &z_0, &z_i);
         if expected_u_i_x != proof.u_i.x[0] {
             return Err(Error::zkIVCVerificationFail);
         }
diff --git a/folding-schemes/src/folding/protogalaxy/circuits.rs b/folding-schemes/src/folding/protogalaxy/circuits.rs
index 74a7332..4ec2fad 100644
--- a/folding-schemes/src/folding/protogalaxy/circuits.rs
+++ b/folding-schemes/src/folding/protogalaxy/circuits.rs
@@ -24,13 +24,16 @@ use super::{
     CommittedInstance, CommittedInstanceVar, ProtoGalaxyCycleFoldConfig,
 };
 use crate::{
-    folding::circuits::{
-        cyclefold::{
-            CycleFoldChallengeGadget, CycleFoldCommittedInstance, CycleFoldCommittedInstanceVar,
-            CycleFoldConfig, NIFSFullGadget,
+    folding::{
+        circuits::{
+            cyclefold::{
+                CycleFoldChallengeGadget, CycleFoldCommittedInstance,
+                CycleFoldCommittedInstanceVar, CycleFoldConfig, NIFSFullGadget,
+            },
+            nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
+            CF1, CF2,
         },
-        nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
-        CF1, CF2,
+        traits::CommittedInstanceVarOps,
     },
     frontend::FCircuit,
     transcript::{AbsorbNonNativeGadget, TranscriptVar},
@@ -346,24 +349,12 @@ where
         // `transcript` is for challenge generation.
         let mut transcript = sponge.clone();
 
-        // get z_{i+1} from the F circuit
-        let i_usize = self.i_usize;
-        let z_i1 =
-            self.F
-                .generate_step_constraints(cs.clone(), i_usize, z_i.clone(), external_inputs)?;
-
         let is_basecase = i.is_zero()?;
 
         // Primary Part
         // P.1. Compute u_i.x
         // u_i.x[0] = H(i, z_0, z_i, U_i)
-        let (u_i_x, _) = U_i.clone().hash(
-            &sponge,
-            pp_hash.clone(),
-            i.clone(),
-            z_0.clone(),
-            z_i.clone(),
-        )?;
+        let (u_i_x, _) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
         // u_i.x[1] = H(cf_U_i)
         let (cf_u_i_x, _) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
 
@@ -380,21 +371,27 @@ where
         )?;
 
         // P.4.a compute and check the first output of F'
+
+        // get z_{i+1} from the F circuit
+        let z_i1 =
+            self.F
+                .generate_step_constraints(cs.clone(), self.i_usize, z_i, external_inputs)?;
+
         // Base case: u_{i+1}.x[0] == H((i+1, z_0, z_{i+1}, U_{\bot})
         // Non-base case: u_{i+1}.x[0] == H((i+1, z_0, z_{i+1}, U_{i+1})
         let (u_i1_x, _) = U_i1.clone().hash(
             &sponge,
-            pp_hash.clone(),
-            i + FpVar::<CF1<C1>>::one(),
-            z_0.clone(),
-            z_i1.clone(),
+            &pp_hash,
+            &(i + FpVar::<CF1<C1>>::one()),
+            &z_0,
+            &z_i1,
         )?;
         let (u_i1_x_base, _) = CommittedInstanceVar::new_constant(cs.clone(), u_dummy)?.hash(
             &sponge,
-            pp_hash.clone(),
-            FpVar::<CF1<C1>>::one(),
-            z_0.clone(),
-            z_i1.clone(),
+            &pp_hash,
+            &FpVar::<CF1<C1>>::one(),
+            &z_0,
+            &z_i1,
         )?;
         let x = FpVar::new_input(cs.clone(), || Ok(self.x.unwrap_or(u_i1_x_base.value()?)))?;
         x.enforce_equal(&is_basecase.select(&u_i1_x_base, &u_i1_x)?)?;
diff --git a/folding-schemes/src/folding/protogalaxy/mod.rs b/folding-schemes/src/folding/protogalaxy/mod.rs
index 9ee44de..37b2836 100644
--- a/folding-schemes/src/folding/protogalaxy/mod.rs
+++ b/folding-schemes/src/folding/protogalaxy/mod.rs
@@ -1,14 +1,14 @@
 /// Implements the scheme described in [ProtoGalaxy](https://eprint.iacr.org/2023/1106.pdf)
 use ark_crypto_primitives::sponge::{
-    constraints::{AbsorbGadget, CryptographicSpongeVar},
-    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
+    poseidon::{PoseidonConfig, PoseidonSponge},
     Absorb, CryptographicSponge,
 };
 use ark_ec::{CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
-    fields::fp::FpVar,
+    eq::EqGadget,
+    fields::{fp::FpVar, FieldVar},
     groups::{CurveVar, GroupOpsBounds},
     R1CSVar, ToConstraintFieldGadget,
 };
@@ -44,6 +44,8 @@ pub(crate) mod utils;
 use circuits::AugmentedFCircuit;
 use folding::Folding;
 
+use super::traits::{CommittedInstanceOps, CommittedInstanceVarOps, WitnessOps, WitnessVarOps};
+
 /// Configuration for ProtoGalaxy's CycleFold circuit
 pub struct ProtoGalaxyCycleFoldConfig<C: CurveGroup> {
     _c: PhantomData<C>,
@@ -83,30 +85,15 @@ impl<C: CurveGroup> CommittedInstance<C> {
     }
 }
 
-impl<C: CurveGroup> CommittedInstance<C>
-where
-    C::ScalarField: Absorb,
-    C::BaseField: PrimeField,
-{
-    /// hash implements the committed instance hash compatible with the gadget implemented in
-    /// CommittedInstanceVar.hash.
-    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U_i` is the
-    /// `CommittedInstance`.
-    pub fn hash(
-        &self,
-        sponge: &PoseidonSponge<C::ScalarField>,
-        pp_hash: C::ScalarField,
-        i: C::ScalarField,
-        z_0: Vec<C::ScalarField>,
-        z_i: Vec<C::ScalarField>,
-    ) -> C::ScalarField {
-        let mut sponge = sponge.clone();
-        sponge.absorb(&pp_hash);
-        sponge.absorb(&i);
-        sponge.absorb(&z_0);
-        sponge.absorb(&z_i);
-        sponge.absorb(&self);
-        sponge.squeeze_field_elements(1)[0]
+impl<C: CurveGroup> CommittedInstanceOps<C> for CommittedInstance<C> {
+    type Var = CommittedInstanceVar<C>;
+
+    fn get_commitments(&self) -> Vec<C> {
+        vec![self.phi]
+    }
+
+    fn is_incoming(&self) -> bool {
+        self.e == Zero::zero() && self.betas.is_empty()
     }
 }
 
@@ -164,34 +151,29 @@ impl<C: CurveGroup> R1CSVar<C::ScalarField> for CommittedInstanceVar<C> {
     }
 }
 
-impl<C: CurveGroup> CommittedInstanceVar<C>
-where
-    C::ScalarField: Absorb,
-    C::BaseField: PrimeField,
-{
-    /// hash implements the committed instance hash compatible with the native implementation from
-    /// CommittedInstance.hash.
-    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U` is the
-    /// `CommittedInstance`.
-    /// Additionally it returns the vector of the field elements from the self parameters, so they
-    /// can be reused in other gadgets avoiding recalculating (reconstraining) them.
-    #[allow(clippy::type_complexity)]
-    pub fn hash(
-        self,
-        sponge: &PoseidonSpongeVar<CF1<C>>,
-        pp_hash: FpVar<CF1<C>>,
-        i: FpVar<CF1<C>>,
-        z_0: Vec<FpVar<CF1<C>>>,
-        z_i: Vec<FpVar<CF1<C>>>,
-    ) -> Result<(FpVar<CF1<C>>, Vec<FpVar<CF1<C>>>), SynthesisError> {
-        let mut sponge = sponge.clone();
-        let U_vec = self.to_sponge_field_elements()?;
-        sponge.absorb(&pp_hash)?;
-        sponge.absorb(&i)?;
-        sponge.absorb(&z_0)?;
-        sponge.absorb(&z_i)?;
-        sponge.absorb(&U_vec)?;
-        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
+impl<C: CurveGroup> CommittedInstanceVarOps<C> for CommittedInstanceVar<C> {
+    type PointVar = NonNativeAffineVar<C>;
+
+    fn get_commitments(&self) -> Vec<Self::PointVar> {
+        vec![self.phi.clone()]
+    }
+
+    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>] {
+        &self.x
+    }
+
+    fn enforce_incoming(&self) -> Result<(), SynthesisError> {
+        if self.betas.is_empty() {
+            self.e.enforce_equal(&FpVar::zero())
+        } else {
+            Err(SynthesisError::Unsatisfiable)
+        }
+    }
+
+    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.betas.enforce_equal(&other.betas)?;
+        self.e.enforce_equal(&other.e)?;
+        self.x.enforce_equal(&other.x)
     }
 }
 
@@ -224,6 +206,44 @@ impl<F: PrimeField> Witness<F> {
     }
 }
 
+impl<F: PrimeField> WitnessOps<F> for Witness<F> {
+    type Var = WitnessVar<F>;
+
+    fn get_openings(&self) -> Vec<(&[F], F)> {
+        vec![(&self.w, self.r_w)]
+    }
+}
+
+/// In-circuit representation of the Witness associated to the CommittedInstance.
+#[derive(Debug, Clone)]
+pub struct WitnessVar<F: PrimeField> {
+    pub W: Vec<FpVar<F>>,
+    pub rW: FpVar<F>,
+}
+
+impl<F: PrimeField> AllocVar<Witness<F>, F> for WitnessVar<F> {
+    fn new_variable<T: Borrow<Witness<F>>>(
+        cs: impl Into<Namespace<F>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        f().and_then(|val| {
+            let cs = cs.into();
+
+            let W = Vec::new_variable(cs.clone(), || Ok(val.borrow().w.to_vec()), mode)?;
+            let rW = FpVar::new_variable(cs.clone(), || Ok(val.borrow().r_w), mode)?;
+
+            Ok(Self { W, rW })
+        })
+    }
+}
+
+impl<F: PrimeField> WitnessVarOps<F> for WitnessVar<F> {
+    fn get_openings(&self) -> Vec<(&[FpVar<F>], FpVar<F>)> {
+        vec![(&self.W, self.rW.clone())]
+    }
+}
+
 #[derive(Debug, thiserror::Error, PartialEq)]
 pub enum ProtoGalaxyError {
     #[error("The remainder from G(X)-F(α)*L_0(X)) / Z(X) should be zero")]
@@ -636,8 +656,8 @@ where
                 &sponge,
                 self.pp_hash,
                 self.i + C1::ScalarField::one(),
-                self.z_0.clone(),
-                z_i1.clone(),
+                &self.z_0,
+                &z_i1,
             );
             // `cf_U_{i+1}` (i.e., `cf_U_1`) is fixed to `cf_U_dummy`, so we
             // just use `self.cf_U_i = cf_U_0 = cf_U_dummy`.
@@ -744,8 +764,8 @@ where
                 &sponge,
                 self.pp_hash,
                 self.i + C1::ScalarField::one(),
-                self.z_0.clone(),
-                z_i1.clone(),
+                &self.z_0,
+                &z_i1,
             );
             cf_u_i1_x = cf_U_i1.hash_cyclefold(&sponge, self.pp_hash);
 
@@ -874,7 +894,7 @@ where
 
         // check that u_i's output points to the running instance
         // u_i.X[0] == H(i, z_0, z_i, U_i)
-        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, z_0, z_i.clone());
+        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, &z_0, &z_i);
         if expected_u_i_x != u_i.x[0] {
             return Err(Error::IVCVerificationFail);
         }
diff --git a/folding-schemes/src/folding/protogalaxy/traits.rs b/folding-schemes/src/folding/protogalaxy/traits.rs
index d735f6e..d67f79e 100644
--- a/folding-schemes/src/folding/protogalaxy/traits.rs
+++ b/folding-schemes/src/folding/protogalaxy/traits.rs
@@ -18,8 +18,8 @@ impl<C: CurveGroup> Absorb for CommittedInstance<C>
 where
     C::ScalarField: Absorb,
 {
-    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
-        unimplemented!()
+    fn to_sponge_bytes(&self, dest: &mut Vec<u8>) {
+        C::ScalarField::batch_to_sponge_bytes(&self.to_sponge_field_elements_as_vec(), dest);
     }
 
     fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
@@ -35,7 +35,7 @@ where
 // Implements the trait for absorbing ProtoGalaxy's CommittedInstanceVar in-circuit.
 impl<C: CurveGroup> AbsorbGadget<C::ScalarField> for CommittedInstanceVar<C> {
     fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
-        unimplemented!()
+        FpVar::batch_to_sponge_bytes(&self.to_sponge_field_elements()?)
     }
 
     fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
@@ -114,3 +114,46 @@ impl<C: CurveGroup> RelaxedR1CS<C, Witness<C::ScalarField>, CommittedInstance<C>
         unimplemented!()
     }
 }
+
+#[cfg(test)]
+pub mod tests {
+    use super::*;
+    use ark_bn254::{Fr, G1Projective as Projective};
+    use ark_r1cs_std::{alloc::AllocVar, R1CSVar};
+    use ark_relations::r1cs::ConstraintSystem;
+    use ark_std::UniformRand;
+    use rand::Rng;
+
+    /// test that checks the native CommittedInstance.to_sponge_{bytes,field_elements}
+    /// vs the R1CS constraints version
+    #[test]
+    pub fn test_committed_instance_to_sponge_preimage() {
+        let mut rng = ark_std::test_rng();
+
+        let t = rng.gen::<u8>() as usize;
+        let io_len = rng.gen::<u8>() as usize;
+
+        let ci = CommittedInstance::<Projective> {
+            phi: Projective::rand(&mut rng),
+            betas: (0..t).map(|_| Fr::rand(&mut rng)).collect(),
+            e: Fr::rand(&mut rng),
+            x: (0..io_len).map(|_| Fr::rand(&mut rng)).collect(),
+        };
+
+        let bytes = ci.to_sponge_bytes_as_vec();
+        let field_elements = ci.to_sponge_field_elements_as_vec();
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        let ciVar =
+            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
+        let bytes_var = ciVar.to_sponge_bytes().unwrap();
+        let field_elements_var = ciVar.to_sponge_field_elements().unwrap();
+
+        assert!(cs.is_satisfied().unwrap());
+
+        // check that the natively computed and in-circuit computed hashes match
+        assert_eq!(bytes_var.value().unwrap(), bytes);
+        assert_eq!(field_elements_var.value().unwrap(), field_elements);
+    }
+}
diff --git a/folding-schemes/src/folding/traits.rs b/folding-schemes/src/folding/traits.rs
new file mode 100644
index 0000000..5815397
--- /dev/null
+++ b/folding-schemes/src/folding/traits.rs
@@ -0,0 +1,121 @@
+use ark_crypto_primitives::sponge::{
+    constraints::{AbsorbGadget, CryptographicSpongeVar},
+    poseidon::constraints::PoseidonSpongeVar,
+    Absorb,
+};
+use ark_ec::CurveGroup;
+use ark_ff::PrimeField;
+use ark_r1cs_std::{alloc::AllocVar, fields::fp::FpVar, ToConstraintFieldGadget};
+use ark_relations::r1cs::SynthesisError;
+
+use crate::{transcript::Transcript, Error};
+
+use super::circuits::CF1;
+
+pub trait CommittedInstanceOps<C: CurveGroup> {
+    /// The in-circuit representation of the committed instance.
+    type Var: AllocVar<Self, CF1<C>> + CommittedInstanceVarOps<C>;
+    /// `hash` implements the committed instance hash compatible with the
+    /// in-circuit implementation from `CommittedInstanceVarOps::hash`.
+    ///
+    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and
+    /// `U_i` is the committed instance `self`.
+    fn hash<T: Transcript<CF1<C>>>(
+        &self,
+        sponge: &T,
+        pp_hash: CF1<C>, // public params hash
+        i: CF1<C>,
+        z_0: &[CF1<C>],
+        z_i: &[CF1<C>],
+    ) -> CF1<C>
+    where
+        CF1<C>: Absorb,
+        Self: Sized + Absorb,
+    {
+        let mut sponge = sponge.clone();
+        sponge.absorb(&pp_hash);
+        sponge.absorb(&i);
+        sponge.absorb(&z_0);
+        sponge.absorb(&z_i);
+        sponge.absorb(&self);
+        sponge.squeeze_field_elements(1)[0]
+    }
+
+    /// Returns the commitments contained in the committed instance.
+    fn get_commitments(&self) -> Vec<C>;
+
+    /// Returns `true` if the committed instance is an incoming instance, and
+    /// `false` if it is a running instance.
+    fn is_incoming(&self) -> bool;
+
+    /// Checks if the committed instance is an incoming instance.
+    fn check_incoming(&self) -> Result<(), Error> {
+        self.is_incoming()
+            .then_some(())
+            .ok_or(Error::NotIncomingCommittedInstance)
+    }
+}
+
+pub trait CommittedInstanceVarOps<C: CurveGroup> {
+    type PointVar: ToConstraintFieldGadget<CF1<C>>;
+    /// `hash` implements the in-circuit committed instance hash compatible with
+    /// the native implementation from `CommittedInstanceOps::hash`.
+    /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and
+    /// `U_i` is the committed instance `self`.
+    ///
+    /// Additionally it returns the in-circuit representation of the committed
+    /// instance `self` as a vector of field elements, so they can be reused in
+    /// other gadgets avoiding recalculating (reconstraining) them.
+    #[allow(clippy::type_complexity)]
+    fn hash(
+        &self,
+        sponge: &PoseidonSpongeVar<CF1<C>>,
+        pp_hash: &FpVar<CF1<C>>,
+        i: &FpVar<CF1<C>>,
+        z_0: &[FpVar<CF1<C>>],
+        z_i: &[FpVar<CF1<C>>],
+    ) -> Result<(FpVar<CF1<C>>, Vec<FpVar<CF1<C>>>), SynthesisError>
+    where
+        Self: AbsorbGadget<CF1<C>>,
+    {
+        let mut sponge = sponge.clone();
+        let U_vec = self.to_sponge_field_elements()?;
+        sponge.absorb(&pp_hash)?;
+        sponge.absorb(&i)?;
+        sponge.absorb(&z_0)?;
+        sponge.absorb(&z_i)?;
+        sponge.absorb(&U_vec)?;
+        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
+    }
+
+    /// Returns the commitments contained in the committed instance.
+    fn get_commitments(&self) -> Vec<Self::PointVar>;
+
+    /// Returns the public inputs contained in the committed instance.
+    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>];
+
+    /// Generates constraints to enforce that the committed instance is an
+    /// incoming instance.
+    fn enforce_incoming(&self) -> Result<(), SynthesisError>;
+
+    /// Generates constraints to enforce that the committed instance `self` is
+    /// partially equal to another committed instance `other`.
+    /// Here, only field elements are compared, while commitments (points) are
+    /// not.
+    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError>;
+}
+
+pub trait WitnessOps<F: PrimeField> {
+    /// The in-circuit representation of the witness.
+    type Var: AllocVar<Self, F> + WitnessVarOps<F>;
+
+    /// Returns the openings (i.e., the values being committed to and the
+    /// randomness) contained in the witness.
+    fn get_openings(&self) -> Vec<(&[F], F)>;
+}
+
+pub trait WitnessVarOps<F: PrimeField> {
+    /// Returns the openings (i.e., the values being committed to and the
+    /// randomness) contained in the witness.
+    fn get_openings(&self) -> Vec<(&[FpVar<F>], FpVar<F>)>;
+}
diff --git a/folding-schemes/src/lib.rs b/folding-schemes/src/lib.rs
index e4b22ef..6ed0805 100644
--- a/folding-schemes/src/lib.rs
+++ b/folding-schemes/src/lib.rs
@@ -43,6 +43,8 @@ pub enum Error {
     IVCVerificationFail,
     #[error("zkIVC verification failed")]
     zkIVCVerificationFail,
+    #[error("Committed instance is expected to be an incoming (fresh) instance")]
+    NotIncomingCommittedInstance,
     #[error("R1CS instance is expected to not be relaxed")]
     R1CSUnrelaxedFail,
     #[error("Could not find the inner ConstraintSystem")]