# PH (complexity)

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

**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}** =

**P**(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

^{PP}**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 **P** ≠ **NP**, since it is only necessary to separate **P** from the more general class **PH**.

## References

- 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. - 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. - https://www.quantamagazine.org/finally-a-problem-that-only-quantum-computers-will-ever-be-able-to-solve-20180621/
- 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.

## General references

- Bürgisser, Peter (2000).
*Completeness and reduction in algebraic complexity theory*. Algorithms and Computation in Mathematics.**7**. Berlin: Springer-Verlag. p. 66. ISBN 3-540-66752-0. Zbl 0948.68082. *Complexity Zoo*: PH