# PH (complexity)

In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy:

${\displaystyle \mathrm {PH} =\bigcup _{k\in \mathbb {N} }\Delta _{k}^{\mathrm {P} }}$

PH was first defined by Larry Stockmeyer.[1] It is a special case of hierarchy of bounded alternating Turing machine. It is contained in P#P = PPP (by Toda's theorem; the class of problems that are decidable by a polynomial time Turing machine with access to a #P or equivalently PP oracle), and also in PSPACE.

PH has a simple logical characterization: it is the set of languages expressible by second-order logic.

PH contains almost all well-known complexity classes inside PSPACE; in particular, it contains P, NP, and co-NP. It even contains probabilistic classes such as BPP and RP. However, there is some evidence that BQP, the class of problems solvable in polynomial time by a quantum computer, is not contained in PH.[2][3]

P = NP if and only if P = PH.[4] This may simplify a potential proof of PNP, since it is only necessary to separate P from the more general class PH.

## References

1. ^ Stockmeyer, Larry J. (1977). "The polynomial-time hierarchy". Theor. Comput. Sci. 3: 1–22. doi:10.1016/0304-3975(76)90061-X. Zbl 0353.02024.
2. ^ Aaronson, Scott (2009). "BQP and the Polynomial Hierarchy". Proc. 42nd Symposium on Theory of Computing (STOC 2009). Association for Computing Machinery. pp. 141–150. arXiv:0910.4698. doi:10.1145/1806689.1806711. ECCC TR09-104.
3. ^ https://www.quantamagazine.org/finally-a-problem-that-only-quantum-computers-will-ever-be-able-to-solve-20180621/
4. ^ Hemaspaandra, Lane (2018). "17.5 Complexity classes". In Rosen, Kenneth H. (ed.). Handbook of Discrete and Combinatorial Mathematics. Discrete Mathematics and Its Applications (2nd ed.). CRC Press. pp. 1308–1314. ISBN 9781351644051.