Circuit Lower Bounds for Merlin–Arthur Classes

Abstract

We show that for each $k>0$, $\mathsf{MA}/1$ ($\mathsf{MA}$ with 1 bit of advice) does not have circuits of size $n^k$. This implies the first superlinear circuit lower bounds for the promise versions of the classes $\mathsf{MA}$, $\mathsf{AM}$, and $\mathsf{ZPP}_{\parallel}^{\mathsf{NP}}$. We extend our main result in several ways. For each $k$, we give an explicit language in $(\mathsf{MA}\cap\mathsf{coMA})/1$ which does not have circuits of size $n^k$. We also adapt our lower bound to the average-case setting; i.e., we show that $\mathsf{MA}/1$ cannot be solved on more than $1/2+1/n^k$ fraction of inputs of length $n$ by circuits of size $n^k$. Furthermore, we prove that $\mathsf{MA}$ does not have arithmetic circuits of size $n^k$ for any $k$. As a corollary to our main result, we obtain that derandomization of $\mathsf{MA}/O(1)$ implies the existence of pseudorandom generators computable using $O(1)$ bits of advice.