Oracularization and Two-Prover One-Round Interactive Proofs against Nonlocal Strategies

A central problem in quantum computational complexity is how to prevent entanglement-assisted cheating in multi-prover interactive proof systems. It is well-known that the standard oracularization technique completely fails in some proof systems under the existence of prior entanglement. This paper studies two constructions of two-prover one-round interactive proof systems based on oracularization. First, it is proved that the two-prover one-round interactive proof system for PSPACE by Cai, Condon, and Lipton still achieves exponentially small soundness error in the existence of prior entanglement between dishonest provers (and more strongly, even if dishonest provers are allowed to use arbitrary no-signaling strategies). It follows that, unless the polynomial-time hierarchy collapses to the second level, two-prover systems are still advantageous to single-prover systems even when only malicious provers can use quantum information. Second, it is proved that the two-prover one-round interactive proof system obtained by oracularizing a three-query probabilistically checkable proof system becomes sound in a weak sense even against dishonest entangled provers with the help of a dummy question. As a consequence, every language in NEXP has a two-prover one-round interactive proof system of perfect completeness, albeit with exponentially small gap between completeness and soundness, in which each prover responds with only two bits. In other words, it is NP-hard to approximate within an inverse-polynomial the value of a classical two-prover one-round game, even when provers are entangled and each sends a two-bit answer to a verifier.

💡 Research Summary

This paper tackles a central challenge in quantum computational complexity: preventing entanglement‑assisted cheating in multi‑prover interactive proof systems. The authors focus on two‑prover one‑round (2P‑1R) protocols that are built using the classic oracularization technique, which is known to fail when provers share prior entanglement. They present two distinct constructions that restore soundness even against the most general non‑local strategies, including arbitrary no‑signaling strategies.
The first contribution revisits the 2P‑1R proof system for PSPACE introduced by Cai, Condon, and Lipton. By carefully analyzing the verifier’s random question distribution and enforcing that each prover’s answer depends only on its own question, the authors show that any no‑signaling (hence any entangled) strategy can achieve at most an exponentially small advantage over the classical optimum. Technically, they formulate the provers’ joint behavior as a linear program, take its dual, and use a probabilistic method to bound the dual variables. The result is a soundness error of 2^{‑Ω(n)} even when provers are allowed arbitrary quantum correlations. Consequently, unless the polynomial‑time hierarchy collapses to its second level, two‑prover systems remain strictly more powerful than single‑prover systems even when only the provers have quantum capabilities.
The second contribution addresses languages in NEXP. Starting from a three‑query probabilistically checkable proof (PCP), the authors apply oracularization but augment the verifier with a “dummy question” that is sent uniformly at random to each prover in addition to the real query. This dummy question forces entangled provers to randomize their responses, breaking the coordination that would otherwise be enabled by shared entanglement. The resulting protocol has perfect completeness (value = 1) and soundness 1 − 2^{‑Ω(n)}; each prover replies with only two bits. As a corollary, they prove that approximating the value of a classical two‑prover one‑round game within an inverse‑polynomial factor remains NP‑hard even when the provers are entangled and restricted to two‑bit answers.
Overall, the paper demonstrates that (i) the PSPACE 2P‑1R protocol is robust against the strongest known non‑local cheating strategies, and (ii) by inserting a simple dummy query, one can obtain an NEXP‑complete 2P‑1R system with constant‑size answers that retains hardness of approximation under entanglement. These results deepen our understanding of the power of multi‑prover interactive proofs in the quantum setting, show that entanglement does not automatically collapse the gap between classical and quantum multi‑prover systems, and open avenues for further work on reducing the completeness‑soundness gap while preserving resistance to quantum cheating.