Complexity Class $\Sigma_3^{\mathsf{P}}$

The decision problems of the form $\exists y_1 \forall y_2 \exists y_3\, W(x, y_1, y_2, y_3)$, where the $y_i$ have size polynomial in $|x|$ and $W$ is decidable in polynomial time. Equivalently, the problems solvable in $\mathsf{NP}$ with a $\Sigma_2^{\mathsf{P}}$ oracle.