Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth

We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φn}, where Φn has size at most p(n) and depth O(1), such that Φn computes the n × n permanent. A circuit family {Φn} is succinct if there exists a nonuniform Boolean circuit family {Cn} with O(logn) many inputs and size n such that that Cn can correctly answer direct connection language queries about Φn succinctness is a relaxation of uniformity. To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this we obtain the lower bound by explicitly constructing a hitting set against arithmetic circuits in the polynomial hierarchy.