Quantum certificate complexity

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)²Q₀(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)=Ω(n/log n).