Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?

Quantum Merlin-Arthur proof systems are a weak form of quantum interactive proof systems, where mighty Merlin as a prover presents a proof in a pure quantum state and Arthur as a verifier performs polynomial-time quantum computation to verify its correctness with high success probability. For a more general treatment, this paper considers quantum “multiple-Merlin”-Arthur proof systems in which Arthur uses multiple quantum proofs unentangled each other for his verification. Although classical multi-proof systems are easily shown to be essentially equivalent to classical single-proof systems, it is unclear whether quantum multi-proof systems collapse to quantum single-proof systems. This paper investigates the possibility that quantum multi-proof systems collapse to quantum single-proof systems, and shows that (i) a necessary and sufficient condition under which the number of quantum proofs is reducible to two and (ii) using multiple quantum proofs does not increase the power of quantum Merlin-Arthur proof systems in the case of perfect soundness. Our proof for the latter result also gives a new characterization of the class NQP, which bridges two existing concepts of “quantum nondeterminism”. It is also shown that (iii) there is a relativized world in which co-NP (actually co-UP) does not have quantum Merlin-Arthur proof systems even with multiple quantum proofs.