Quantum and reversible verification of proofs using constant memory space

Abstract

Non-interactive verification of proofs or certificates by deterministic  verifiers in polynomial time with mighty provers is used to characterize languages in NP. We initiate the study of the computational complexity of similar non-interactive proof-verification procedures by quantum and reversible verifiers who are permitted to use only a constant amount of memory storage. By modeling those weak verifiers as quantum and reversible finite automata, we investigate fundamental properties of such non-interactive proof systems and demonstrate that languages admitting proof systems in which verifiers must scan the input in real time are exactly regular languages. On the contrary, when we allow verifiers to move their tape heads in all directions, the corresponding proof systems are empowered to recognize non-stochastic, non-contextfree, and NP-complete languages.