  The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such
that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$
satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$.
We prove that many spectral lower bounds for the chromatic number, $\chi$, are
also lower bounds for $\xi$. This result complements a previous result by the
authors, in which they showed that spectral lower bounds for $\chi$ are also
lower bounds for the quantum chromatic number $\chi_q$. It is known that the
quantum chromatic number and the orthogonal rank are incomparable.
  We conclude by proving an inertial lower bound for the projective rank
$\xi_f$, and conjecture that a stronger inertial lower bound for $\xi$ is also
a lower bound for $\xi_f$.
