Certificate Games and Consequences for the Classical Adversary Bound

We introduce and study Certificate Game complexity, a measure of complexity based on the probability of winning a game where two players are given inputs with different function values and are asked to output some index $i$ such that $x_i\neq y_i$, in a zero-communication setting. We study four versions of certificate games, namely private coin, public coin, shared entanglement and non-signaling games. The public-coin variant of certificate games gives a new characterization of the classical adversary bound, a lower bound on randomized query complexity which was introduced as a classical version of the quantum (non-negative) quantum adversary bound. We show that complexity in the public coin model (therefore also the classical adversary) is bounded above by certificate complexity, as well as by expectational certificate complexity and sabotage complexity. On the other hand, it is bounded below by fractional and randomized certificate complexity. The quantum measure reveals an interesting and surprising difference between classical and quantum query models: the quantum certificate game complexity can be quadratically larger than quantum query complexity. We use non-signaling, a notion from quantum information, to give a lower bound of $n$ on the quantum certificate game complexity of the OR function, whose quantum query complexity is $\Theta(\sqrt{n})$, then go on to show that this ``non-signaling bottleneck'' applies to all functions with high sensitivity, block sensitivity, fractional block sensitivity, as well as classical adversary. This implies the collapse of all models of certificate games, except private randomness, to the classical adversary bound. We consider the single-bit version of certificate games, where the inputs of the two players are restricted to having Hamming distance 1, and give a new characterization of sensitivity and spectral sensitivity.