Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation R0(f) = O(Q2(f)^2 Q0(f) log n) for total f, where R0, Q2, and Q0 are zero-error randomized, bounded-error quantum, and zero-error quantum query complexities respectively. Finally we give asymptotic gaps between the measures, including a total f for which C(f) is superquadratic in QC(f), and a symmetric partial f for which QC(f) = O(1) yet Q2(f) = Omega(n log n).
It is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N. Nisan, SIAM J. Comput.20, No. 6 (1991), 999?1007). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree, we address structural properties of Arthur?Merlin games in this model and prove some lower bounds. We consider two cases of interest, the first when the length of communication between the players is limited and the second, if it is not. While in the first case we can carry over the relations between the corresponding Turing complexity classes, in the second case we observe in contrast with Turing complexity that a one-round Merlin?Arthur protocol is as powerful as a general interactive proof system and, in particular, can simulate a one-round Arthur?Merlin protocol. Moreover, we show that sometimes a Merlin?Arthur protocol can be more efficient than an Arthur?Merlin protocol and than a Merlin?Arthur protocol with limited communication. This is the case for a boolean function whose set of zeroes is a code with high minimum distance and a natural uniformity condition. Such functions provide an example when the Merlin?Arthur complexity is 1 with one-sided error ??(23, 1), but at the same time the nondeterministic decision tree complexity is ?(n). The latter should be contrasted with another fact we prove. Namely, if a function has Merlin?Arthur complexity 1 with one-sided error probability ??(0, 23, then its nondeterministic complexity is bounded by a constant. Other results of the paper include connections with the block sensitivity and related combinatorial properties of a boolean function.