Strong average-case lower bounds from non-trivial derandomization

We prove that for all constants a, NQP = NTIME[n polylog(n)] cannot be (1/2 + 2−log a n )-approximated by 2log a n -size ACC 0 ∘ THR circuits ( ACC 0 circuits with a bottom layer of THR gates). Previously, it was even open whether E NP can be (1/2+1/√n)-approximated by AC 0[⊕] circuits. As a straightforward application, we obtain an infinitely often ( NE ∩ coNE)/1-computable pseudorandom generator for poly-size ACC 0 circuits with seed length 2logє n , for all є > 0.