The plural in the title hints that there are several complexities involved here.
The most traditional one is the computational complexity of the verifier; but, in the context of Interactive Proofs of Proximity (IPPs), the query complexity of the verifier and the communication complexity of the system should ... more >>>

In a $k$-XOR game $\Psi$ over $\mathbb{F}_p$, a verifier samples challenges $(x_1,\ldots,x_k)\sim\mu$ where $\mu$ is a distribution over $\Sigma_1\times\ldots\times \Sigma_k$, sends $x_i$ to player $i$ and receives an answer $f_i(x_i)\in\mathbb{F}_p$, and accepts if and only if $f_1(x_1)+\ldots+f_k(x_k)=t(x_1,\ldots,x_k)$, where $t\colon \Sigma_1\times\ldots\times \Sigma_k\to\mathbb{F}_p$ is a fixed map. We show that if $\Psi$ ... more >>>

The Poisson approximation theorem states that when $m$ balls are tossed uniformly and randomly into $n$ bins, the distribution of the bin loads can sometimes be approximated by $n$ independent Poisson random variables of mean $m/n$. We show that when the balls are tossed via a random $\mathbb{F}_2$-linear map, ... more >>>