The levels of the polynomial hierarchy are defined by logics. Level 0 is polynomial time: $\Sigma_0^p$ is PTIME and $\Pi_0^p$ is coPTIME. For $k \ge 1$, $\Sigma_k^p$ is the class of decision problems definable by a second-order sentence with $k$ alternating blocks of second-order quantifiers, the first existential, over a first-order kernel, and $\Pi_k^p$ the class of those definable with the first block universal; these are the levels of the hierarchy by the theorems of Fagin and Stockmeyer. In particular $\Sigma_1^p$ is NP, and coNP is $\Pi_1^p$, the problems definable in universal second-order logic. Hardness for each level is the cofinal hardness of the NP core, under relativized ordered first-order reductions.

PH is the union of the levels: a problem is in PH when it is in $\Sigma_k^p$ for some $k$, and PH-hard when it is hard for every level.