  The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally
defined as the minimal number of colors necessary in a quantum protocol in
which two provers that cannot communicate with each other but share an
entangled state can convince an interrogator with certainty that they have a
coloring of the graph. We use an equivalent purely combinatorial definition of
$\chi_q(G)$ to prove that many spectral lower bounds for the chromatic number,
$\chi(G)$, are also lower bounds for $\chi_q(G)$. This is achieved using
techniques from linear algebra called pinching and twirling. We illustrate our
results with some examples.
