The paper "Batch Proofs Are Statistically Hiding" by Nir Bitansky et al. explores the existence and limitations of batch proofs and arguments in computational complexity theory. Batch proofs are interactive proof systems that enable a prover to convince a verifier that multiple inputs $x_1, \ldots, x_t$ belong to a language $\mathcal{L} \in \text{NP}$ with significantly less communication than sending individual witnesses. The authors present two main results: 1. **Statistically Sound Batch Proofs**: They show that if there exists a statistically sound batch proof for a language $\mathcal{L} \in \text{NP}$, then $\mathcal{L}$ must have a statistically witness-indistinguishable (SWI) proof with inverse polynomial error and a non-uniform honest prover. This implies that achieving batch proofs beyond UP (where witness indistinguishability is trivial) is challenging, as it would require SWI proofs for all of NP, which is currently unknown. 2. **Computationally Sound Batch Proofs**: They demonstrate that if there exist computationally sound batch proofs (or batch arguments, BARGs) for NP, and assuming one-way functions, these can imply statistical zero-knowledge (SZK) arguments for NP with similar round complexity and error rates. This suggests that constant-round interactive BARGs from one-way functions could yield constant-round SZK arguments, which is surprising given the current understanding of SZK arguments. The authors also provide a transformation that converts any batch protocol into a protocol for a single instance with statistical hiding properties, achieving statistical witness indistinguishability (SWI) against an honest verifier. This transformation has implications for both statistical and computational soundness, with caveats on the error rates and the nature of the honest prover strategy. In the non-interactive setting, the authors further show how to construct explicit (uniform) protocols with negligible error rates, leading to new non-interactive zero-knowledge (NIZK) arguments. They achieve this by leveraging the notion of "somewhere soundness" in non-interactive BARGs and using techniques such as dual-mode commitments and lossy encryption. Overall, the paper contributes to the understanding of the limitations and possibilities of batch proofs and arguments, providing both negative and positive results that have implications for the design of efficient and secure cryptographic protocols.The paper "Batch Proofs Are Statistically Hiding" by Nir Bitansky et al. explores the existence and limitations of batch proofs and arguments in computational complexity theory. Batch proofs are interactive proof systems that enable a prover to convince a verifier that multiple inputs $x_1, \ldots, x_t$ belong to a language $\mathcal{L} \in \text{NP}$ with significantly less communication than sending individual witnesses. The authors present two main results: 1. **Statistically Sound Batch Proofs**: They show that if there exists a statistically sound batch proof for a language $\mathcal{L} \in \text{NP}$, then $\mathcal{L}$ must have a statistically witness-indistinguishable (SWI) proof with inverse polynomial error and a non-uniform honest prover. This implies that achieving batch proofs beyond UP (where witness indistinguishability is trivial) is challenging, as it would require SWI proofs for all of NP, which is currently unknown. 2. **Computationally Sound Batch Proofs**: They demonstrate that if there exist computationally sound batch proofs (or batch arguments, BARGs) for NP, and assuming one-way functions, these can imply statistical zero-knowledge (SZK) arguments for NP with similar round complexity and error rates. This suggests that constant-round interactive BARGs from one-way functions could yield constant-round SZK arguments, which is surprising given the current understanding of SZK arguments. The authors also provide a transformation that converts any batch protocol into a protocol for a single instance with statistical hiding properties, achieving statistical witness indistinguishability (SWI) against an honest verifier. This transformation has implications for both statistical and computational soundness, with caveats on the error rates and the nature of the honest prover strategy. In the non-interactive setting, the authors further show how to construct explicit (uniform) protocols with negligible error rates, leading to new non-interactive zero-knowledge (NIZK) arguments. They achieve this by leveraging the notion of "somewhere soundness" in non-interactive BARGs and using techniques such as dual-mode commitments and lossy encryption. Overall, the paper contributes to the understanding of the limitations and possibilities of batch proofs and arguments, providing both negative and positive results that have implications for the design of efficient and secure cryptographic protocols.