Toda's theorem is a result in computational complexity theory that was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy"[1] and was given the 1998 Gödel Prize.
The theorem states that the entire polynomial hierarchy PH is contained in PPP; this implies a closely related statement, that PH is contained in P#P.
#P is the problem of exactly counting the number of solutions to a polynomially-verifiable question (that is, to a question in NP), while loosely speaking, PP is the problem of giving an answer that is correct more than half the time.
The class P#P consists of all the problems that can be solved in polynomial time if you have access to instantaneous answers to any counting problem in #P (polynomial time relative to a #P oracle).
[2] An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real Turing machines) was proved by Saugata Basu and Thierry Zell in 2009[3] and a complex analogue of Toda's theorem was proved by Saugata Basu in 2011.