Abstract: We prove that for all constants a, NQP = NTIME[npolylog(n)] cannot be (1/2 + 2−loga n)-approximated by 2loga n-size ACC0 ∘ THR circuits ( ACC0 circuits with a bottom layer of THR gates). Previously, it was even open whether E NP can be (1/2+1/√n)-approximated by AC0[⊕] circuits. As a straightforward application, we obtain an infinitely often ( NE ∩ coNE)/1-computable pseudorandom generator for poly-size ACC0 circuits with seed length 2logєn, for all є > 0. More generally, we establish a connection showing that, for a typical circuit class C, non-trivial nondeterministic algorithms estimating the acceptance probability of a given S-size C circuit with an additive error 1/S (we call it a CAPP algorithm) imply strong (1/2 + 1/nω(1)) average-case lower bounds for nondeterministic time classes against C circuits. Note that the existence of such (deterministic) algorithms is much weaker than the widely believed conjecture PromiseBPP = PromiseP. We also apply our results to several sub-classes of TC0 circuits. First, we show that for all k, NP cannot be (1/2 + n−k)-approximated by nk-size Sum∘ THR circuits (exact ℝ-linear combination of threshold gates), improving the corresponding worst-case result in [Williams, CCC 2018]. Second, we establish strong average-case lower bounds and build ( NE ∩ coNE)/1-computable PRGs for Sum ∘ PTF circuits, for various regimes of degrees. Third, we show that non-trivial CAPP algorithms for MAJ ∘ MAJ indeed already imply worst-case lower bounds for TC30 ( MAJ ∘ MAJ ∘ MAJ). Since exponential lower bounds for MAJ ∘ MAJ are already known, this suggests TC30 lower bounds are probably within reach. Our new results build on a line of recent works, including [Murray and Williams, STOC 2018], [Chen and Williams, CCC 2019], and [Chen, FOCS 2019]. In particular, it strengthens the corresponding (1/2 + 1/polylog(n))-inapproximability average-case lower bounds in [Chen, FOCS 2019]. The two important technical ingredients are techniques from Cryptography in NC0 [Applebaum et al., SICOMP 2006], and Probabilistic Checkable Proofs of Proximity with NC1-computable proofs.