Complexity Class $\Sigma_2^{\mathsf{P}}$

The decision problems of the form $\exists y_1 \forall y_2\, W(x, y_1, y_2)$, where $y_1, y_2$ have size polynomial in $|x|$ and $W$ is decidable in polynomial time. Equivalently, the problems solvable in $\mathsf{NP}$ with an $\mathsf{NP}$ oracle, $\Sigma_2^{\mathsf{P}} = \mathsf{NP}^{\mathsf{NP}}$. It forms the second level of the polynomial hierarchy together with $\Pi_2^{\mathsf{P}}$.