In a $3$-XOR game $\mathcal{G}$, the verifier samples a challenge $(x,y,z)\sim \mu$ where $\mu$ is a probability distribution over $\Sigma\times\Gamma\times\Phi$, and a map $t\colon \Sigma\times\Gamma\times\Phi\to\mathcal{A}$ for a finite Abelian group $\mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$ and $z$ to the players Alice, Bob and Charlie ... more >>>

The notion of closure of a set of linear forms, first introduced by Efremenko, Garlik, and Itsykson [EGI-STOC-24], has proven instrumental in proving lower bounds on the sizes of regular and bounded-depth Res($\oplus)$ refutations [EGI-STOC-24, AI-STOC-25]. In this work, we present amortized closure, an enhancement that retains the properties of ... more >>>

Proving complexity lower bounds remains a challenging task: currently, we only know how to prove conditional uniform (algorithm) lower bounds and nonuniform (circuit) lower bounds in restricted circuit models. About a decade ago, Williams (STOC 2010) showed how to derive nonuniform lower bounds from uniform upper bounds: roughly, by designing ... more >>>