Eric Allender

A very recent paper by Caussinus, McKenzie, Therien, and Vollmer

[CMTV] shows that ACC^0 is properly contained in ModPH, and TC^0

is properly contained in the counting hierarchy. Thus, [CMTV] shows

that there are problems in ModPH that require superpolynomial-size

uniform ACC^0 ...
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Cody Murray, Ryan Williams

We prove that if every problem in $NP$ has $n^k$-size circuits for a fixed constant $k$, then for every $NP$-verifier and every yes-instance $x$ of length $n$ for that verifier, the verifier's search space has an $n^{O(k^3)}$-size witness circuit: a witness for $x$ that can be encoded with a circuit ... more >>>