Improved Circuit Lower Bounds and Quantum-Classical Separations

We continue the study of the circuit class GC^0, which augments AC^0 with unbounded-fan-in gates that compute arbitrary functions inside a sufficiently small Hamming ball but must be constant outside it. While GC^0 can compute functions requiring exponential-size circuits, Kumar (CCC 2023) showed that switching-lemma lower bounds for AC^0 extend to GC^0 with no loss in parameters. We prove a parallel result for the polynomial method: any lower bound for AC^0[p] obtained via the polynomial method extends to GC^0[p] without loss in parameters. As a consequence, we show that the majority function MAJ requires depth-$d$ GC^0[p] circuits of size $2^{\Omega(n^{1/2(d-1)})}$, matching the best-known lower bounds for AC^0[p]. This yields the most expressive class of non-monotone circuits for which exponential-size lower bounds are known for an explicit function. We also prove a similar result for the algorithmic method, showing that E^NP requires exponential-size GCC^0 circuits, extending a result of Williams (JACM 2014). Finally, leveraging our improved classical lower bounds, we establish the strongest known unconditional separations between quantum and classical circuit classes. We separate QNC^0 from GC^0 and GC^0[p] in various settings and show that BQLOGTIME is not contained in GC^0. As a consequence, we construct an oracle relative to which BQP lies outside uniform GC^0, extending the Raz-Tal oracle separation between BQP and PH (STOC 2019).