Certificate Games and Consequences for the Classical Adversary Bound

TL;DR

This work introduces certificate games as a game-based lens on the classical adversary bound and its quantum variants. It defines four zero-communication models, links their winning probabilities to key complexity measures, and establishes tight equivalences among them (notably $CG^{pub}$, $CG^{ns}$, $CG^{*}$, and $CMM$) for total and partial functions. The authors develop hashing- and LP-based techniques to obtain sharp upper and lower bounds, uncovering a non-signaling bottleneck that constrains quantum strategies. They derive strong partial-function separations (e.g., ApInd) and give single-bit characterizations that connect certificate games to sensitivity and spectral sensitivity. Overall, the paper unifies several complexity measures under the certificate-game framework and highlights open questions around composition and the sensitivity-block-sensitivity relationship.

Abstract

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 $Θ(\sqrt{n})$, 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.

Figures (1)

• Figure 1: Some known relations among complexity measures for total functions. An arrow from $A$ to $B$ indicates that for every total Boolean function $f$, $B(f) = O(A(f))$. Double arrows indicate results in this paper, and boxes indicate new complexity measures. Single arrows indicate known results and references are omitted from the diagram for space considerations. Most references can be found in the tables in yu19ABKRT21 and we cite others in later sections. Known relations about $\mathsf{EC}$ are given in JKKLSSV20, and $\mathsf{FC} = O((\mathsf{MM})^2)$ is proven in ABK21. Fractional certificate complexity $\mathsf{FC}$ is equal to fractional block sensitivity and to randomized certificate complexity RC (up to multiplicative constants). $\mathsf{MM}$ is the minimax formulation of the positive adversary method. $\mathsf{MM} = O(\mathsf{CMM})$ is proved in KT16.

Theorems & Definitions (88)

• Definition 1.1: Certificate game
• Definition 2.1: Non-signaling strategy
• Proposition 2.2: Characterization of non-signaling strategies
• Definition 2.3: Private coin certificate game complexity
• Definition 2.4: Public coin certificate game complexity
• Definition 2.5: Shared entanglement certificate game complexity
• Definition 2.6: Non-signaling certificate game complexity
• Proposition 2.7
• Proposition 3.1: Dual formulation of $\mathsf{CG}^{\text{\upshape pub}}$
• Proposition 3.2: Dual formulation of $\mathsf{CG}^\mathrm{ns}$