We prove that the XOR of $k$ majorities on disjoint blocks of $\ell$ bits has correlation at most $(4d/\sqrt{\ell})^k$ with every Boolean-valued degree-$d$ polynomial over an arbitrary field. By known techniques, we obtain explicit pseudorandom generators that fool such polynomials on $n$ Boolean inputs with seed length $d^2\cdot\polylog(n/\varepsilon)$. The correlation and the generator extend to small $\mathsf{AC}^0[q]$ circuits, for every fixed prime power $q$. These results resolve several long-standing questions in circuit complexity and pseudorandomness.

翻译：暂无翻译