About $\Sigma_2^p$

Does $\Sigma_2^p$ have a complete problem?

$L = \{x\mid \exists y_1 \forall y_2 M(x, y_1, y_2)\} \in \Sigma_2^p$

All the machines we have been using are Deterministic Machine.

Alternating Turing Machine For $\mathsf{HAMCIRCUIT}$