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.

Generalized the main results to all fields, added new applications and improved some existing ones, and expanded the exposition.

We prove that the XOR of $k$ majorities on disjoint blocks of \(\ell\) bits has correlation at most \((2d/\sqrt{\ell})^k\) with every degree-\(d\) polynomial over \(\mathbb F_2\). By known techniques, this implies pseudorandom generators with polylogarithmic seed length for low-degree polynomials over $\mathbb F_2$ and for alternating circuits with parity gates.