In this paper, we study a weak version of quantum nondeterministic communication complexity, corresponding to the most natural generalization of classical nondeterminism, in which a classical proof has to be checked with probability one by a quantum protocol. Another stronger definition of quantum nondeterminism has already been extensively studied, corresponding to the view of quantum nondeterminism as unbounded-error one-sided quantum computation, but, although being mathematically

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... this definition fundamentally lacks the original view of nondeterministic processes as proof-checking processes. In this paper, we prove that, in the framework of communication complexity, even the weak version of quantum nondeterminism is strictly stronger than classical nondeterminism. More precisely, we show the first separation, for a total function, of quantum weakly nondeterministic and classical nondeterministic communication complexity. This separation is quadratic and shows than classical proofs can be checked more efficiently by quantum protocols than by classical ones, in the framework of communication complexity.