Algebraic methods and bounded formulas. Selected references Merrick Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. Seinosuke Toda and Mitsunori Ogiwara.
Counting classes are at least as hard as the polynomial-time hierarchy.
L. G. Valiant and V. V. Vazirani. NP is as easy as detecting unique solutions. Andrew Chi-Chih Yao.