Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries

Abstract

We prove a near-maximum ($2^n/n$) circuit lower bound for the complexity class ${\mathsf{E}}^{\mathrm{pr}\mathsf{MA}}{/}_1$, corresponding to exponential time with access to a promise-$\mathsf{MA}$ oracle and one bit of advice. Our proof incorporates the iterative win-win paradigm (Chen--Lu--Oliveira--Ren--Santhanam, FOCS'23), the reduction from the Range Avoidance problem to circuit lower bounds (Jeřábek, Ann. Pure Appl. Log. '04; Korten, FOCS'21), and the PCP theorem. Crucial to our proof is the analysis of the complexity class $\mathsf{P}^\mathsf{NP}[{\textsf{#rounds}}=r, {\textsf{length}}=s]$, which is ${\mathsf{P}}^{\mathsf{NP}}$ with $r(n)$ adaptive rounds of $\mathsf{NP}$ queries, where each $\mathsf{NP}$ query has witness length $s(n)$.