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

• Published in: 2009 24th Annual IEEE Conference on Computational Complexity pp. 217 - 228
• Main Authors: Ito, T., Kobayashi, H., Matsumoto, K.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.07.2009
• Subjects: Computational complexity; Computer science; Cryptography; Entanglement; Game theory; Indium tin oxide; Informatics; Multi-prover interactive proof systems; Polynomials; Power system modeling; Quantum computing; Quantum entanglement; Quantum nonlocality
• ISBN: 9780769537177; 0769537170
• ISSN: 1093-0159
• DOI: 10.1109/CCC.2009.22

This paper presents three results on the power of two-prover one-round interactive proof systems based on oracularization under the existence of prior entanglement between dishonest provers. It is proved that the two-prover one-round interactive proof system for PSPACE by Cai, Condon, and Lipton [JCSS 48:183-193, 1994] 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. It is also shown that a "dummy" question may be helpful when constructing an entanglement-resistant multi-prover system via oracularization. This affirmatively settles a question posed by Kempe et al. [FOCS 2008, pp. 447-456] and every language in NEXP is proved to have a two-prover one-round interactive proof system even against entangled provers, albeit with exponentially small gap between completeness and soundness. In other words, it is NP-hard to approximate within an inverse-polynomial the value of a classical two-prover one-round game against entangled provers. Finally, both for the above proof system for NEXP and for the quantum two-prover one-round proof system for NEXP proposed by Kempe et al., it is proved that exponentially small completeness-soundness gaps are best achievable unless soundness analysis uses the structure of the underlying system with unentangled provers.