@@ -1,6 +1,6 @@
 [package]
 name = "nova-snark"
-version = "0.19.0"
+version = "0.19.1"
 authors = ["Srinath Setty <srinath@microsoft.com>"]
 edition = "2021"
 description = "Recursive zkSNARKs without trusted setup"

@@ -1095,8 +1095,8 @@ assert_eq!(zn_secondary, zn_secondary_direct);
     assert_eq!(zn_secondary, vec![<G2 as Group>::Scalar::from(2460515u64)]);
 
     // run the compressed snark with Spark compiler
-    type CC1Prime = spartan::spark::SparkEngine<G1, EE1>;
-    type CC2Prime = spartan::spark::SparkEngine<G2, EE2>;
+    type CC1Prime = spartan::spark::SparkEngine<G1>;
+    type CC2Prime = spartan::spark::SparkEngine<G2>;
     type S1Prime = spartan::RelaxedR1CSSNARK<G1, EE1, CC1Prime>;
     type S2Prime = spartan::RelaxedR1CSSNARK<G2, EE2, CC2Prime>;
 

@@ -12,7 +12,7 @@ traits::{
     evaluation::EvaluationEngineTrait, snark::RelaxedR1CSSNARKTrait, Group, TranscriptEngineTrait,
     TranscriptReprTrait,
   },
-  CommitmentKey,
+  Commitment, CommitmentKey,
 };
 use ff::Field;
 use itertools::concat;
@@ -21,8 +21,67 @@ use rayon::prelude::*;
 use serde::{Deserialize, Serialize};
 use sumcheck::SumcheckProof;
 
+/// A type that holds a witness to a polynomial evaluation instance
+#[allow(dead_code)]
+pub struct PolyEvalWitness<G: Group> {
+  p: Vec<G::Scalar>, // polynomial
+}
+
+impl<G: Group> PolyEvalWitness<G> {
+  fn pad(W: &[PolyEvalWitness<G>]) -> Vec<PolyEvalWitness<G>> {
+    // determine the maximum size
+    if let Some(n) = W.iter().map(|w| w.p.len()).max() {
+      W.iter()
+        .map(|w| {
+          let mut p = w.p.clone();
+          p.resize(n, G::Scalar::zero());
+          PolyEvalWitness { p }
+        })
+        .collect()
+    } else {
+      Vec::new()
+    }
+  }
+
+  fn weighted_sum(W: &[PolyEvalWitness<G>], s: &[G::Scalar]) -> PolyEvalWitness<G> {
+    assert_eq!(W.len(), s.len());
+    let mut p = vec![G::Scalar::zero(); W[0].p.len()];
+    for i in 0..W.len() {
+      for j in 0..W[i].p.len() {
+        p[j] += W[i].p[j] * s[i]
+      }
+    }
+    PolyEvalWitness { p }
+  }
+}
+
+/// A type that holds a polynomial evaluation instance
+#[allow(dead_code)]
+pub struct PolyEvalInstance<G: Group> {
+  c: Commitment<G>,  // commitment to the polynomial
+  x: Vec<G::Scalar>, // evaluation point
+  e: G::Scalar,      // claimed evaluation
+}
+
+impl<G: Group> PolyEvalInstance<G> {
+  fn pad(U: &[PolyEvalInstance<G>]) -> Vec<PolyEvalInstance<G>> {
+    // determine the maximum size
+    if let Some(ell) = U.iter().map(|u| u.x.len()).max() {
+      U.iter()
+        .map(|u| {
+          let mut x = vec![G::Scalar::zero(); ell - u.x.len()];
+          x.extend(u.x.clone());
+          PolyEvalInstance { c: u.c, x, e: u.e }
+        })
+        .collect()
+    } else {
+      Vec::new()
+    }
+  }
+}
+
 /// A trait that defines the behavior of a computation commitment engine
-pub trait CompCommitmentEngineTrait<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
+pub trait CompCommitmentEngineTrait<G: Group> {
   /// A type that holds opening hint
   type Decommitment: Clone + Send + Sync + Serialize + for<'de> Deserialize<'de>;
 
@@ -46,22 +105,26 @@
   /// proves an evaluation of R1CS matrices viewed as polynomials
   fn prove(
     ck: &CommitmentKey<G>,
-    ek: &EE::ProverKey,
     S: &R1CSShape<G>,
     decomm: &Self::Decommitment,
     comm: &Self::Commitment,
     r: &(&[G::Scalar], &[G::Scalar]),
     transcript: &mut G::TE,
-  ) -> Result<Self::EvaluationArgument, NovaError>;
+  ) -> Result<
+    (
+      Self::EvaluationArgument,
+      Vec<(PolyEvalWitness<G>, PolyEvalInstance<G>)>,
+    ),
+    NovaError,
+  >;
 
   /// verifies an evaluation of R1CS matrices viewed as polynomials and returns verified evaluations
   fn verify(
-    vk: &EE::VerifierKey,
     comm: &Self::Commitment,
     r: &(&[G::Scalar], &[G::Scalar]),
     arg: &Self::EvaluationArgument,
     transcript: &mut G::TE,
-  ) -> Result<(G::Scalar, G::Scalar, G::Scalar), NovaError>;
+  ) -> Result<(G::Scalar, G::Scalar, G::Scalar, Vec<PolyEvalInstance<G>>), NovaError>;
 }
 
 /// A type that represents the prover's key
@@ -70,7 +133,7 @@ #[serde(bound = "")]
 pub struct ProverKey<
   G: Group,
   EE: EvaluationEngineTrait<G, CE = G::CE>,
-  CC: CompCommitmentEngineTrait<G, EE>,
+  CC: CompCommitmentEngineTrait<G>,
 > {
   pk_ee: EE::ProverKey,
   S: R1CSShape<G>,
@@ -84,7 +147,7 @@ #[serde(bound = "")]
 pub struct VerifierKey<
   G: Group,
   EE: EvaluationEngineTrait<G, CE = G::CE>,
-  CC: CompCommitmentEngineTrait<G, EE>,
+  CC: CompCommitmentEngineTrait<G>,
 > {
   num_cons: usize,
   num_vars: usize,
@@ -100,21 +163,20 @@ #[serde(bound = "")]
 pub struct RelaxedR1CSSNARK<
   G: Group,
   EE: EvaluationEngineTrait<G, CE = G::CE>,
-  CC: CompCommitmentEngineTrait<G, EE>,
+  CC: CompCommitmentEngineTrait<G>,
 > {
   sc_proof_outer: SumcheckProof<G>,
   claims_outer: (G::Scalar, G::Scalar, G::Scalar),
   eval_E: G::Scalar,
   sc_proof_inner: SumcheckProof<G>,
   eval_W: G::Scalar,
+  eval_arg_cc: CC::EvaluationArgument,
   sc_proof_batch: SumcheckProof<G>,
-  eval_E_prime: G::Scalar,
-  eval_W_prime: G::Scalar,
+  evals_batch: Vec<G::Scalar>,
   eval_arg: EE::EvaluationArgument,
-  eval_arg_cc: CC::EvaluationArgument,
 }
 
-impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>, CC: CompCommitmentEngineTrait<G, EE>>
+impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>, CC: CompCommitmentEngineTrait<G>>
   RelaxedR1CSSNARKTrait<G> for RelaxedR1CSSNARK<G, EE, CC>
 {
   type ProverKey = ProverKey<G, EE, CC>;
@@ -292,9 +354,8 @@ &mut transcript,
     )?;
 
     // we now prove evaluations of R1CS matrices at (r_x, r_y)
-    let eval_arg_cc = CC::prove(
+    let (eval_arg_cc, mut w_u_vec) = CC::prove(
       ck,
-      &pk.pk_ee,
       &pk.S,
       &pk.decomm,
       &pk.comm,
@@ -302,52 +363,111 @@ &(&r_x, &r_y),
       &mut transcript,
     )?;
 
-    let eval_W = MultilinearPolynomial::new(W.W.clone()).evaluate(&r_y[1..]);
-    transcript.absorb(b"eval_W", &eval_W);
+    // add additional claims about W and E polynomials to the list from CC
+    let eval_W = MultilinearPolynomial::evaluate_with(&W.W, &r_y[1..]);
+    w_u_vec.push((
+      PolyEvalWitness { p: W.W.clone() },
+      PolyEvalInstance {
+        c: U.comm_W,
+        x: r_y[1..].to_vec(),
+        e: eval_W,
+      },
+    ));
 
-    // We will now reduce eval_W =? W(r_y[1..]) and eval_W =? E(r_x) into
+    w_u_vec.push((
+      PolyEvalWitness { p: W.E },
+      PolyEvalInstance {
+        c: U.comm_E,
+        x: r_x,
+        e: eval_E,
+      },
+    ));
+
+    // We will now reduce a vector of claims of evaluations at different points into claims about them at the same point.
+    // For example, eval_W =? W(r_y[1..]) and eval_W =? E(r_x) into
     // two claims: eval_W_prime =? W(rz) and eval_E_prime =? E(rz)
     // We can them combine the two into one: eval_W_prime + gamma * eval_E_prime =? (W + gamma*E)(rz),
     // where gamma is a public challenge
     // Since commitments to W and E are homomorphic, the verifier can compute a commitment
     // to the batched polynomial.
-    let rho = transcript.squeeze(b"rho")?;
+    assert!(w_u_vec.len() >= 2);
 
-    let claim_batch_joint = eval_E + rho * eval_W;
-    let num_rounds_z = num_rounds_x;
-    let comb_func =
-      |poly_A_comp: &G::Scalar,
-       poly_B_comp: &G::Scalar,
-       poly_C_comp: &G::Scalar,
-       poly_D_comp: &G::Scalar|
-       -> G::Scalar { *poly_A_comp * *poly_B_comp + rho * *poly_C_comp * *poly_D_comp };
-    let (sc_proof_batch, r_z, claims_batch) = SumcheckProof::prove_quad_sum(
+    let (w_vec, u_vec): (Vec<PolyEvalWitness<G>>, Vec<PolyEvalInstance<G>>) =
+      w_u_vec.into_iter().unzip();
+    let w_vec_padded = PolyEvalWitness::pad(&w_vec); // pad the polynomials to be of the same size
+    let u_vec_padded = PolyEvalInstance::pad(&u_vec); // pad the evaluation points
+
+    let powers = |s: &G::Scalar, n: usize| -> Vec<G::Scalar> {
+      assert!(n >= 1);
+      let mut powers = Vec::new();
+      powers.push(G::Scalar::one());
+      for i in 1..n {
+        powers.push(powers[i - 1] * s);
+      }
+      powers
+    };
+
+    // generate a challenge
+    let rho = transcript.squeeze(b"r")?;
+    let num_claims = w_vec_padded.len();
+    let powers_of_rho = powers(&rho, num_claims);
+    let claim_batch_joint = u_vec_padded
+      .iter()
+      .zip(powers_of_rho.iter())
+      .map(|(u, p)| u.e * p)
+      .fold(G::Scalar::zero(), |acc, item| acc + item);
+
+    let mut polys_left: Vec<MultilinearPolynomial<G::Scalar>> = w_vec_padded
+      .iter()
+      .map(|w| MultilinearPolynomial::new(w.p.clone()))
+      .collect();
+    let mut polys_right: Vec<MultilinearPolynomial<G::Scalar>> = u_vec_padded
+      .iter()
+      .map(|u| MultilinearPolynomial::new(EqPolynomial::new(u.x.clone()).evals()))
+      .collect();
+
+    let num_rounds_z = u_vec_padded[0].x.len();
+    let comb_func = |poly_A_comp: &G::Scalar, poly_B_comp: &G::Scalar| -> G::Scalar {
+      *poly_A_comp * *poly_B_comp
+    };
+    let (sc_proof_batch, r_z, claims_batch) = SumcheckProof::prove_quad_batch(
       &claim_batch_joint,
       num_rounds_z,
-      &mut MultilinearPolynomial::new(EqPolynomial::new(r_x.clone()).evals()),
-      &mut MultilinearPolynomial::new(W.E.clone()),
-      &mut MultilinearPolynomial::new(EqPolynomial::new(r_y[1..].to_vec()).evals()),
-      &mut MultilinearPolynomial::new(W.W.clone()),
+      &mut polys_left,
+      &mut polys_right,
+      &powers_of_rho,
       comb_func,
       &mut transcript,
     )?;
 
-    let eval_E_prime = claims_batch[1];
-    let eval_W_prime = claims_batch[3];
-    transcript.absorb(b"claims_batch", &[eval_E_prime, eval_W_prime].as_slice());
+    let (claims_batch_left, _): (Vec<G::Scalar>, Vec<G::Scalar>) = claims_batch;
+
+    transcript.absorb(b"l", &claims_batch_left.as_slice());
 
     // we now combine evaluation claims at the same point rz into one
-    let gamma = transcript.squeeze(b"gamma")?;
-    let comm = U.comm_E + U.comm_W * gamma;
-    let poly = W
-      .E
+    let gamma = transcript.squeeze(b"g")?;
+    let powers_of_gamma: Vec<G::Scalar> = powers(&gamma, num_claims);
+    let comm_joint = u_vec_padded
+      .iter()
+      .zip(powers_of_gamma.iter())
+      .map(|(u, g_i)| u.c * *g_i)
+      .fold(Commitment::<G>::default(), |acc, item| acc + item);
+    let poly_joint = PolyEvalWitness::weighted_sum(&w_vec_padded, &powers_of_gamma);
+    let eval_joint = claims_batch_left
       .iter()
-      .zip(W.W.iter())
-      .map(|(e, w)| *e + gamma * w)
-      .collect::<Vec<G::Scalar>>();
-    let eval = eval_E_prime + gamma * eval_W_prime;
+      .zip(powers_of_gamma.iter())
+      .map(|(e, g_i)| *e * *g_i)
+      .fold(G::Scalar::zero(), |acc, item| acc + item);
 
-    let eval_arg = EE::prove(ck, &pk.pk_ee, &mut transcript, &comm, &poly, &r_z, &eval)?;
+    let eval_arg = EE::prove(
+      ck,
+      &pk.pk_ee,
+      &mut transcript,
+      &comm_joint,
+      &poly_joint.p,
+      &r_z,
+      &eval_joint,
+    )?;
 
     Ok(RelaxedR1CSSNARK {
       sc_proof_outer,
@@ -355,11 +475,10 @@ claims_outer: (claim_Az, claim_Bz, claim_Cz),
       eval_E,
       sc_proof_inner,
       eval_W,
+      eval_arg_cc,
       sc_proof_batch,
-      eval_E_prime,
-      eval_W_prime,
+      evals_batch: claims_batch_left,
       eval_arg,
-      eval_arg_cc,
     })
   }
 
@@ -433,25 +552,50 @@ (G::Scalar::one() - r_y[0]) * self.eval_W + r_y[0] * eval_X
     };
 
     // verify evaluation argument to retrieve evaluations of R1CS matrices
-    let (eval_A, eval_B, eval_C) = CC::verify(
-      &vk.vk_ee,
-      &vk.comm,
-      &(&r_x, &r_y),
-      &self.eval_arg_cc,
-      &mut transcript,
-    )?;
+    let (eval_A, eval_B, eval_C, mut u_vec) =
+      CC::verify(&vk.comm, &(&r_x, &r_y), &self.eval_arg_cc, &mut transcript)?;
 
     let claim_inner_final_expected = (eval_A + r * eval_B + r * r * eval_C) * eval_Z;
     if claim_inner_final != claim_inner_final_expected {
       return Err(NovaError::InvalidSumcheckProof);
     }
 
-    // batch sum-check
-    transcript.absorb(b"eval_W", &self.eval_W);
+    // add additional claims about W and E polynomials to the list from CC
+    u_vec.push(PolyEvalInstance {
+      c: U.comm_W,
+      x: r_y[1..].to_vec(),
+      e: self.eval_W,
+    });
+
+    u_vec.push(PolyEvalInstance {
+      c: U.comm_E,
+      x: r_x,
+      e: self.eval_E,
+    });
+
+    let u_vec_padded = PolyEvalInstance::pad(&u_vec); // pad the evaluation points
 
-    let rho = transcript.squeeze(b"rho")?;
-    let claim_batch_joint = self.eval_E + rho * self.eval_W;
-    let num_rounds_z = num_rounds_x;
+    let powers = |s: &G::Scalar, n: usize| -> Vec<G::Scalar> {
+      assert!(n >= 1);
+      let mut powers = Vec::new();
+      powers.push(G::Scalar::one());
+      for i in 1..n {
+        powers.push(powers[i - 1] * s);
+      }
+      powers
+    };
+
+    // generate a challenge
+    let rho = transcript.squeeze(b"r")?;
+    let num_claims = u_vec.len();
+    let powers_of_rho = powers(&rho, num_claims);
+    let claim_batch_joint = u_vec
+      .iter()
+      .zip(powers_of_rho.iter())
+      .map(|(u, p)| u.e * p)
+      .fold(G::Scalar::zero(), |acc, item| acc + item);
+
+    let num_rounds_z = u_vec_padded[0].x.len();
     let (claim_batch_final, r_z) =
       self
         .sc_proof_batch
@@ -459,32 +603,47 @@ .verify(claim_batch_joint, num_rounds_z, 2, &mut transcript)?;
 
     let claim_batch_final_expected = {
       let poly_rz = EqPolynomial::new(r_z.clone());
-      let rz_rx = poly_rz.evaluate(&r_x);
-      let rz_ry = poly_rz.evaluate(&r_y[1..]);
-      rz_rx * self.eval_E_prime + rho * rz_ry * self.eval_W_prime
+      let evals = u_vec_padded
+        .iter()
+        .map(|u| poly_rz.evaluate(&u.x))
+        .collect::<Vec<G::Scalar>>();
+
+      evals
+        .iter()
+        .zip(self.evals_batch.iter())
+        .zip(powers_of_rho.iter())
+        .map(|((e_i, p_i), rho_i)| *e_i * *p_i * rho_i)
+        .fold(G::Scalar::zero(), |acc, item| acc + item)
     };
 
     if claim_batch_final != claim_batch_final_expected {
       return Err(NovaError::InvalidSumcheckProof);
     }
 
-    transcript.absorb(
-      b"claims_batch",
-      &[self.eval_E_prime, self.eval_W_prime].as_slice(),
-    );
+    transcript.absorb(b"l", &self.evals_batch.as_slice());
 
     // we now combine evaluation claims at the same point rz into one
-    let gamma = transcript.squeeze(b"gamma")?;
-    let comm = U.comm_E + U.comm_W * gamma;
-    let eval = self.eval_E_prime + gamma * self.eval_W_prime;
+    let gamma = transcript.squeeze(b"g")?;
+    let powers_of_gamma: Vec<G::Scalar> = powers(&gamma, num_claims);
+    let comm_joint = u_vec_padded
+      .iter()
+      .zip(powers_of_gamma.iter())
+      .map(|(u, g_i)| u.c * *g_i)
+      .fold(Commitment::<G>::default(), |acc, item| acc + item);
+    let eval_joint = self
+      .evals_batch
+      .iter()
+      .zip(powers_of_gamma.iter())
+      .map(|(e, g_i)| *e * *g_i)
+      .fold(G::Scalar::zero(), |acc, item| acc + item);
 
-    // verify eval_W and eval_E
+    // verify
     EE::verify(
       &vk.vk_ee,
       &mut transcript,
-      &comm,
+      &comm_joint,
       &r_z,
-      &eval,
+      &eval_joint,
       &self.eval_arg,
     )?;
 

@@ -3,7 +3,7 @@ //! We also provide a trivial implementation that has the verifier evaluate the sparse polynomials
 use crate::{
   errors::NovaError,
   r1cs::R1CSShape,
-  spartan::{math::Math, CompCommitmentEngineTrait},
+  spartan::{math::Math, CompCommitmentEngineTrait, PolyEvalInstance, PolyEvalWitness},
   traits::{evaluation::EvaluationEngineTrait, Group, TranscriptReprTrait},
   CommitmentKey,
 };
@@ -43,7 +43,7 @@ self.S.to_transcript_bytes()
   }
 }
 
-impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> CompCommitmentEngineTrait<G, EE>
+impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> CompCommitmentEngineTrait<G>
   for TrivialCompComputationEngine<G, EE>
 {
   type Decommitment = TrivialDecommitment<G>;
@@ -66,29 +66,36 @@
   /// proves an evaluation of R1CS matrices viewed as polynomials
   fn prove(
     _ck: &CommitmentKey<G>,
-    _ek: &EE::ProverKey,
     _S: &R1CSShape<G>,
     _decomm: &Self::Decommitment,
     _comm: &Self::Commitment,
     _r: &(&[G::Scalar], &[G::Scalar]),
     _transcript: &mut G::TE,
-  ) -> Result<Self::EvaluationArgument, NovaError> {
-    Ok(TrivialEvaluationArgument {
-      _p: Default::default(),
-    })
+  ) -> Result<
+    (
+      Self::EvaluationArgument,
+      Vec<(PolyEvalWitness<G>, PolyEvalInstance<G>)>,
+    ),
+    NovaError,
+  > {
+    Ok((
+      TrivialEvaluationArgument {
+        _p: Default::default(),
+      },
+      Vec::new(),
+    ))
   }
 
   /// verifies an evaluation of R1CS matrices viewed as polynomials
   fn verify(
-    _vk: &EE::VerifierKey,
     comm: &Self::Commitment,
     r: &(&[G::Scalar], &[G::Scalar]),
     _arg: &Self::EvaluationArgument,
     _transcript: &mut G::TE,
-  ) -> Result<(G::Scalar, G::Scalar, G::Scalar), NovaError> {
+  ) -> Result<(G::Scalar, G::Scalar, G::Scalar, Vec<PolyEvalInstance<G>>), NovaError> {
     let (r_x, r_y) = r;
     let evals = SparsePolynomial::<G>::multi_evaluate(&[&comm.S.A, &comm.S.B, &comm.S.C], r_x, r_y);
-    Ok((evals[0], evals[1], evals[2]))
+    Ok((evals[0], evals[1], evals[2], Vec::new()))
   }
 }
 
@@ -98,9 +105,8 @@
 use sparse::{SparseEvaluationArgument, SparsePolynomial, SparsePolynomialCommitment};
 
 /// A non-trivial implementation of `CompCommitmentEngineTrait` using Spartan's SPARK compiler
-pub struct SparkEngine<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
+pub struct SparkEngine<G: Group> {
   _p: PhantomData<G>,
-  _p2: PhantomData<EE>,
 }
 
 /// An implementation of Spark decommitment
@@ -156,18 +162,16 @@
 /// Provides an implementation of a trivial evaluation argument
 #[derive(Clone, Serialize, Deserialize)]
 #[serde(bound = "")]
-pub struct SparkEvaluationArgument<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
-  arg_A: SparseEvaluationArgument<G, EE>,
-  arg_B: SparseEvaluationArgument<G, EE>,
-  arg_C: SparseEvaluationArgument<G, EE>,
+pub struct SparkEvaluationArgument<G: Group> {
+  arg_A: SparseEvaluationArgument<G>,
+  arg_B: SparseEvaluationArgument<G>,
+  arg_C: SparseEvaluationArgument<G>,
 }
 
-impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> CompCommitmentEngineTrait<G, EE>
-  for SparkEngine<G, EE>
-{
+impl<G: Group> CompCommitmentEngineTrait<G> for SparkEngine<G> {
   type Decommitment = SparkDecommitment<G>;
   type Commitment = SparkCommitment<G>;
-  type EvaluationArgument = SparkEvaluationArgument<G, EE>;
+  type EvaluationArgument = SparkEvaluationArgument<G>;
 
   /// commits to R1CS matrices
   fn commit(
@@ -182,39 +186,60 @@
   /// proves an evaluation of R1CS matrices viewed as polynomials
   fn prove(
     ck: &CommitmentKey<G>,
-    pk_ee: &EE::ProverKey,
     S: &R1CSShape<G>,
     decomm: &Self::Decommitment,
     comm: &Self::Commitment,
     r: &(&[G::Scalar], &[G::Scalar]),
     transcript: &mut G::TE,
-  ) -> Result<Self::EvaluationArgument, NovaError> {
-    let arg_A =
-      SparseEvaluationArgument::prove(ck, pk_ee, &decomm.A, &S.A, &comm.comm_A, r, transcript)?;
-    let arg_B =
-      SparseEvaluationArgument::prove(ck, pk_ee, &decomm.B, &S.B, &comm.comm_B, r, transcript)?;
-    let arg_C =
-      SparseEvaluationArgument::prove(ck, pk_ee, &decomm.C, &S.C, &comm.comm_C, r, transcript)?;
+  ) -> Result<
+    (
+      Self::EvaluationArgument,
+      Vec<(PolyEvalWitness<G>, PolyEvalInstance<G>)>,
+    ),
+    NovaError,
+  > {
+    let (arg_A, u_w_vec_A) =
+      SparseEvaluationArgument::prove(ck, &decomm.A, &S.A, &comm.comm_A, r, transcript)?;
+    let (arg_B, u_w_vec_B) =
+      SparseEvaluationArgument::prove(ck, &decomm.B, &S.B, &comm.comm_B, r, transcript)?;
+    let (arg_C, u_w_vec_C) =
+      SparseEvaluationArgument::prove(ck, &decomm.C, &S.C, &comm.comm_C, r, transcript)?;
 
-    Ok(SparkEvaluationArgument {
-      arg_A,
-      arg_B,
-      arg_C,
-    })
+    let u_w_vec = {
+      let mut u_w_vec = u_w_vec_A;
+      u_w_vec.extend(u_w_vec_B);
+      u_w_vec.extend(u_w_vec_C);
+      u_w_vec
+    };
+
+    Ok((
+      SparkEvaluationArgument {
+        arg_A,
+        arg_B,
+        arg_C,
+      },
+      u_w_vec,
+    ))
   }
 
   /// verifies an evaluation of R1CS matrices viewed as polynomials
   fn verify(
-    vk_ee: &EE::VerifierKey,
     comm: &Self::Commitment,
     r: &(&[G::Scalar], &[G::Scalar]),
     arg: &Self::EvaluationArgument,
     transcript: &mut G::TE,
-  ) -> Result<(G::Scalar, G::Scalar, G::Scalar), NovaError> {
-    let eval_A = arg.arg_A.verify(vk_ee, &comm.comm_A, r, transcript)?;
-    let eval_B = arg.arg_B.verify(vk_ee, &comm.comm_B, r, transcript)?;
-    let eval_C = arg.arg_C.verify(vk_ee, &comm.comm_C, r, transcript)?;
+  ) -> Result<(G::Scalar, G::Scalar, G::Scalar, Vec<PolyEvalInstance<G>>), NovaError> {
+    let (eval_A, u_vec_A) = arg.arg_A.verify(&comm.comm_A, r, transcript)?;
+    let (eval_B, u_vec_B) = arg.arg_B.verify(&comm.comm_B, r, transcript)?;
+    let (eval_C, u_vec_C) = arg.arg_C.verify(&comm.comm_C, r, transcript)?;
+
+    let u_vec = {
+      let mut u_vec = u_vec_A;
+      u_vec.extend(u_vec_B);
+      u_vec.extend(u_vec_C);
+      u_vec
+    };
 
-    Ok((eval_A, eval_B, eval_C))
+    Ok((eval_A, eval_B, eval_C, u_vec))
   }
 }

@@ -7,12 +7,9 @@ spartan::{
     math::Math,
     polynomial::{EqPolynomial, MultilinearPolynomial},
     spark::product::{IdentityPolynomial, ProductArgumentBatched},
-    SumcheckProof,
+    PolyEvalInstance, PolyEvalWitness, SumcheckProof,
   },
-  traits::{
-    commitment::CommitmentEngineTrait, evaluation::EvaluationEngineTrait, Group,
-    TranscriptEngineTrait, TranscriptReprTrait,
-  },
+  traits::{commitment::CommitmentEngineTrait, Group, TranscriptEngineTrait, TranscriptReprTrait},
   Commitment, CommitmentKey,
 };
 use ff::Field;
@@ -227,7 +224,7 @@ }
 
 #[derive(Clone, Debug, Serialize, Deserialize)]
 #[serde(bound = "")]
-pub struct SparseEvaluationArgument<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> {
+pub struct SparseEvaluationArgument<G: Group> {
   // claimed evaluation
   eval: G::Scalar,
 
@@ -240,7 +237,6 @@ sc_proof_eval: SumcheckProof<G>,
   eval_E_row: G::Scalar,
   eval_E_col: G::Scalar,
   eval_val: G::Scalar,
-  arg_eval: EE::EvaluationArgument,
 
   // proof that E_row is well-formed
   eval_init_row: G::Scalar,
@@ -262,23 +258,22 @@ eval_col: G::Scalar,
   eval_col_read_ts: G::Scalar,
   eval_E_col2: G::Scalar,
   eval_col_audit_ts: G::Scalar,
-  arg_row_col_joint: EE::EvaluationArgument,
-  arg_row_audit_ts: EE::EvaluationArgument,
-  arg_col_audit_ts: EE::EvaluationArgument,
 }
 
-impl<G: Group, EE: EvaluationEngineTrait<G, CE = G::CE>> SparseEvaluationArgument<G, EE> {
+impl<G: Group> SparseEvaluationArgument<G> {
   pub fn prove(
     ck: &CommitmentKey<G>,
-    pk_ee: &EE::ProverKey,
     poly: &SparsePolynomial<G>,
     sparse: &[(usize, usize, G::Scalar)],
     comm: &SparsePolynomialCommitment<G>,
     r: &(&[G::Scalar], &[G::Scalar]),
     transcript: &mut G::TE,
-  ) -> Result<Self, NovaError> {
+  ) -> Result<(Self, Vec<(PolyEvalWitness<G>, PolyEvalInstance<G>)>), NovaError> {
     let (r_x, r_y) = r;
     let eval = SparsePolynomial::<G>::multi_evaluate(&[sparse], r_x, r_y)[0];
+
+    // keep track of evaluation claims
+    let mut w_u_vec: Vec<(PolyEvalWitness<G>, PolyEvalInstance<G>)> = Vec::new();
 
     // compute oracles to prove the correctness of `eval`
     let (E_row, E_col, T_x, T_y) = SparsePolynomial::<G>::evaluation_oracles(sparse, r_x, r_y);
@@ -316,15 +311,16 @@ .zip(E_col.iter())
       .zip(val.iter())
       .map(|((a, b), c)| *a + rho * *b + rho * rho * *c)
       .collect::<Vec<G::Scalar>>();
-    let arg_eval = EE::prove(
-      ck,
-      pk_ee,
-      transcript,
-      &comm_joint,
-      &poly_eval,
-      &r_eval,
-      &eval_joint,
-    )?;
+
+    // add the claim to prove for later
+    w_u_vec.push((
+      PolyEvalWitness { p: poly_eval },
+      PolyEvalInstance {
+        c: comm_joint,
+        x: r_eval,
+        e: eval_joint,
+      },
+    ));
 
     // we now need to prove that E_row and E_col are well-formed
     // we use memory checking: H(INIT) * H(WS) =? H(RS) * H(FINAL)
@@ -462,37 +458,41 @@ *x + c * y + c * c * z + c * c * c * m + c * c * c * c * n + c * c * c * c * c * q
       })
       .collect::<Vec<_>>();
 
-    let arg_row_col_joint = EE::prove(
-      ck,
-      pk_ee,
-      transcript,
-      &comm_joint,
-      &poly_joint,
-      &r_read_write_row_col,
-      &eval_joint,
-    )?;
+    // add the claim to prove for later
+    w_u_vec.push((
+      PolyEvalWitness { p: poly_joint },
+      PolyEvalInstance {
+        c: comm_joint,
+        x: r_read_write_row_col,
+        e: eval_joint,
+      },
+    ));
 
-    let arg_row_audit_ts = EE::prove(
-      ck,
-      pk_ee,
-      transcript,
-      &comm.comm_row_audit_ts,
-      &poly.row_audit_ts,
-      &r_init_audit_row,
-      &eval_row_audit_ts,
-    )?;
+    transcript.absorb(b"a", &eval_row_audit_ts); // add evaluation to transcript, commitment is already in
+    w_u_vec.push((
+      PolyEvalWitness {
+        p: poly.row_audit_ts.clone(),
+      },
+      PolyEvalInstance {
+        c: comm.comm_row_audit_ts,
+        x: r_init_audit_row,
+        e: eval_row_audit_ts,
+      },
+    ));
 
-    let arg_col_audit_ts = EE::prove(
-      ck,
-      pk_ee,
-      transcript,
-      &comm.comm_col_audit_ts,
-      &poly.col_audit_ts,
-      &r_init_audit_col,
-      &eval_col_audit_ts,
-    )?;
+    transcript.absorb(b"a", &eval_col_audit_ts); // add evaluation to transcript, commitment is already in
+    w_u_vec.push((
+      PolyEvalWitness {
+        p: poly.col_audit_ts.clone(),
+      },
+      PolyEvalInstance {
+        c: comm.comm_col_audit_ts,
+        x: r_init_audit_col,
+        e: eval_col_audit_ts,
+      },
+    ));
 
-    Ok(Self {
+    let eval_arg = Self {
       // claimed evaluation
       eval,
 
@@ -505,7 +505,6 @@ sc_proof_eval,
       eval_E_row: claims_eval[0],
       eval_E_col: claims_eval[1],
       eval_val: claims_eval[2],
-      arg_eval,
 
       // proof that E_row and E_row are well-formed
       eval_init_row: eval_init_audit_row[0],
@@ -527,20 +526,21 @@ eval_col,
       eval_col_read_ts,
       eval_E_col2,
       eval_col_audit_ts,
-      arg_row_col_joint,
-      arg_row_audit_ts,
-      arg_col_audit_ts,
-    })
+    };
+
+    Ok((eval_arg, w_u_vec))
   }
 
   pub fn verify(
     &self,
-    vk_ee: &EE::VerifierKey,
     comm: &SparsePolynomialCommitment<G>,
     r: &(&[G::Scalar], &[G::Scalar]),
     transcript: &mut G::TE,
-  ) -> Result<G::Scalar, NovaError> {
+  ) -> Result<(G::Scalar, Vec<PolyEvalInstance<G>>), NovaError> {
     let (r_x, r_y) = r;
+
+    // keep track of evaluation claims
+    let mut u_vec: Vec<PolyEvalInstance<G>> = Vec::new();
 
     // append the transcript and scalar
     transcript.absorb(b"E", &vec![self.comm_E_row, self.comm_E_col].as_slice());
@@ -562,14 +562,13 @@ // prove evaluations of E_row, E_col and val at r_eval
     let rho = transcript.squeeze(b"r")?;
     let comm_joint = self.comm_E_row + self.comm_E_col * rho + comm.comm_val * rho * rho;
     let eval_joint = self.eval_E_row + rho * self.eval_E_col + rho * rho * self.eval_val;
-    EE::verify(
-      vk_ee,
-      transcript,
-      &comm_joint,
-      &r_eval,
-      &eval_joint,
-      &self.arg_eval,
-    )?;
+
+    // add the claim to prove for later
+    u_vec.push(PolyEvalInstance {
+      c: comm_joint,
+      x: r_eval,
+      e: eval_joint,
+    });
 
     // (2) verify if E_row and E_col are well formed
     let gamma_1 = transcript.squeeze(b"g1")?;
@@ -700,33 +699,26 @@ + comm.comm_col * c * c * c
       + comm.comm_col_read_ts * c * c * c * c
       + self.comm_E_col * c * c * c * c * c;
 
-    EE::verify(
-      vk_ee,
-      transcript,
-      &comm_joint,
-      &r_read_write_row_col,
-      &eval_joint,
-      &self.arg_row_col_joint,
-    )?;
+    u_vec.push(PolyEvalInstance {
+      c: comm_joint,
+      x: r_read_write_row_col,
+      e: eval_joint,
+    });
 
-    EE::verify(
-      vk_ee,
-      transcript,
-      &comm.comm_row_audit_ts,
-      &r_init_audit_row,
-      &self.eval_row_audit_ts,
-      &self.arg_row_audit_ts,
-    )?;
+    transcript.absorb(b"a", &self.eval_row_audit_ts); // add evaluation to transcript, commitment is already in
+    u_vec.push(PolyEvalInstance {
+      c: comm.comm_row_audit_ts,
+      x: r_init_audit_row,
+      e: self.eval_row_audit_ts,
+    });
 
-    EE::verify(
-      vk_ee,
-      transcript,
-      &comm.comm_col_audit_ts,
-      &r_init_audit_col,
-      &self.eval_col_audit_ts,
-      &self.arg_col_audit_ts,
-    )?;
+    transcript.absorb(b"a", &self.eval_col_audit_ts); // add evaluation to transcript, commitment is already in
+    u_vec.push(PolyEvalInstance {
+      c: comm.comm_col_audit_ts,
+      x: r_init_audit_col,
+      e: self.eval_col_audit_ts,
+    });
 
-    Ok(self.eval)
+    Ok((self.eval, u_vec))
   }
 }

@@ -126,54 +126,52 @@ vec![poly_A[0], poly_B[0]],
     ))
   }
 
-  pub fn prove_quad_sum<F>(
+  pub fn prove_quad_batch<F>(
     claim: &G::Scalar,
     num_rounds: usize,
-    poly_A: &mut MultilinearPolynomial<G::Scalar>,
-    poly_B: &mut MultilinearPolynomial<G::Scalar>,
-    poly_C: &mut MultilinearPolynomial<G::Scalar>,
-    poly_D: &mut MultilinearPolynomial<G::Scalar>,
+    poly_A_vec: &mut Vec<MultilinearPolynomial<G::Scalar>>,
+    poly_B_vec: &mut Vec<MultilinearPolynomial<G::Scalar>>,
+    coeffs: &[G::Scalar],
     comb_func: F,
     transcript: &mut G::TE,
-  ) -> Result<(Self, Vec<G::Scalar>, Vec<G::Scalar>), NovaError>
+  ) -> Result<(Self, Vec<G::Scalar>, (Vec<G::Scalar>, Vec<G::Scalar>)), NovaError>
   where
-    F: Fn(&G::Scalar, &G::Scalar, &G::Scalar, &G::Scalar) -> G::Scalar + Sync,
+    F: Fn(&G::Scalar, &G::Scalar) -> G::Scalar,
   {
+    let mut e = *claim;
     let mut r: Vec<G::Scalar> = Vec::new();
-    let mut polys: Vec<CompressedUniPoly<G>> = Vec::new();
-    let mut claim_per_round = *claim;
-    for _ in 0..num_rounds {
-      let poly = {
+    let mut quad_polys: Vec<CompressedUniPoly<G>> = Vec::new();
+
+    for _j in 0..num_rounds {
+      let mut evals: Vec<(G::Scalar, G::Scalar)> = Vec::new();
+
+      for (poly_A, poly_B) in poly_A_vec.iter().zip(poly_B_vec.iter()) {
+        let mut eval_point_0 = G::Scalar::zero();
+        let mut eval_point_2 = G::Scalar::zero();
+
         let len = poly_A.len() / 2;
+        for i in 0..len {
+          // eval 0: bound_func is A(low)
+          eval_point_0 += comb_func(&poly_A[i], &poly_B[i]);
 
-        // Make an iterator returning the contributions to the evaluations
-        let (eval_point_0, eval_point_2) = (0..len)
-          .into_par_iter()
-          .map(|i| {
-            // eval 0: bound_func is A(low)
-            let eval_point_0 = comb_func(&poly_A[i], &poly_B[i], &poly_C[i], &poly_D[i]);
+          // eval 2: bound_func is -A(low) + 2*A(high)
+          let poly_A_bound_point = poly_A[len + i] + poly_A[len + i] - poly_A[i];
+          let poly_B_bound_point = poly_B[len + i] + poly_B[len + i] - poly_B[i];
+          eval_point_2 += comb_func(&poly_A_bound_point, &poly_B_bound_point);
+        }
+
+        evals.push((eval_point_0, eval_point_2));
+      }
 
-            // eval 2: bound_func is -A(low) + 2*A(high)
-            let poly_A_bound_point = poly_A[len + i] + poly_A[len + i] - poly_A[i];
-            let poly_B_bound_point = poly_B[len + i] + poly_B[len + i] - poly_B[i];
-            let poly_C_bound_point = poly_C[len + i] + poly_C[len + i] - poly_C[i];
-            let poly_D_bound_point = poly_D[len + i] + poly_D[len + i] - poly_D[i];
-            let eval_point_2 = comb_func(
-              &poly_A_bound_point,
-              &poly_B_bound_point,
-              &poly_C_bound_point,
-              &poly_D_bound_point,
-            );
-            (eval_point_0, eval_point_2)
-          })
-          .reduce(
-            || (G::Scalar::zero(), G::Scalar::zero()),
-            |a, b| (a.0 + b.0, a.1 + b.1),
-          );
+      let evals_combined_0 = (0..evals.len())
+        .map(|i| evals[i].0 * coeffs[i])
+        .fold(G::Scalar::zero(), |acc, item| acc + item);
+      let evals_combined_2 = (0..evals.len())
+        .map(|i| evals[i].1 * coeffs[i])
+        .fold(G::Scalar::zero(), |acc, item| acc + item);
 
-        let evals = vec![eval_point_0, claim_per_round - eval_point_0, eval_point_2];
-        UniPoly::from_evals(&evals)
-      };
+      let evals = vec![evals_combined_0, e - evals_combined_0, evals_combined_2];
+      let poly = UniPoly::from_evals(&evals);
 
       // append the prover's message to the transcript
       transcript.absorb(b"p", &poly);
@@ -181,25 +179,22 @@
       // derive the verifier's challenge for the next round
       let r_i = transcript.squeeze(b"c")?;
       r.push(r_i);
-      polys.push(poly.compress());
 
-      // Set up next round
-      claim_per_round = poly.evaluate(&r_i);
+      // bound all tables to the verifier's challenege
+      for (poly_A, poly_B) in poly_A_vec.iter_mut().zip(poly_B_vec.iter_mut()) {
+        poly_A.bound_poly_var_top(&r_i);
+        poly_B.bound_poly_var_top(&r_i);
+      }
 
-      // bound all tables to the verifier's challenege
-      poly_A.bound_poly_var_top(&r_i);
-      poly_B.bound_poly_var_top(&r_i);
-      poly_C.bound_poly_var_top(&r_i);
-      poly_D.bound_poly_var_top(&r_i);
+      e = poly.evaluate(&r_i);
+      quad_polys.push(poly.compress());
     }
 
-    Ok((
-      SumcheckProof {
-        compressed_polys: polys,
-      },
-      r,
-      vec![poly_A[0], poly_B[0], poly_C[0], poly_D[0]],
-    ))
+    let poly_A_final = (0..poly_A_vec.len()).map(|i| poly_A_vec[i][0]).collect();
+    let poly_B_final = (0..poly_B_vec.len()).map(|i| poly_B_vec[i][0]).collect();
+    let claims_prod = (poly_A_final, poly_B_final);
+
+    Ok((SumcheckProof::new(quad_polys), r, claims_prod))
   }
 
   pub fn prove_cubic_with_additive_term<F>(