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I'm not sure how to answer (1), but (2) is known to imply circuit lower bounds against NQP (non-deterministic quasi-polynomial time). This is from Cody Murray and Ryan Williams' STOC 2018 paper. In fact, they show that these lower bounds follow from faster algorithms for what they call Gap Circuit Unsatisfiability: given a circuit $C$ on $n$ variables and ...


It is perhaps reasonable to think that circuits that people would come up with would come along with easily verifiable proofs of their correctness. (This was indirectly inspired by Dietrich & Wilson's notion of "groups of black box type", and is in the spirit of Hartmanis's "provable complexity.") Towards this end, we consider: Definition: "Certified $...


Don't forget that even though the fan-in is unbounded, the number of gates is polynomially bounded in the number of variables $n$ (in the definition of $\mathsf{AC}$ for instance) .

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