A Simpler Proof of the Worst-Case to Average-Case Reduction for Polynomial Hierarchy via Symmetry of Information.

We give a simplified proof of Hirahara’s result showing that DistPH ⊆ AvgP would imply PH ⊆ DTIME[2 ] [Hir21a]. The argument relies on a proof of the new result: Symmetry of Information for time-bounded Kolmogorov complexity under the assumption that NP is easy on average, which is interesting in its own right and generalizes the “weak” Symmetry of Information theorem from the original [Hir21a].