Let $P \in \mathbb{C}[\tau, \zeta_1, \ldots, \zeta_n]$ be a quadratic polynomial for which the $\tau$-variable is non-characteristic. We characterize when the zero-variety $V(P)$ of $P$ satisfies the Phragmén-Lindelöf condition $PL(\omega)$ or equivalently when the pair $(\mathbb{R}_x^n, \mathbb{R}_\tau \times \mathbb{R}_x^n)$ is of evolution in the class ${\mathcal E}_\omega$ for the partial differential operator $P(D)$ with symbol $P$.