  Compact stars containing quark matter may masquerade as neutron stars in the
range of measured mass and radius, making it difficult to draw firm conclusions
on the phase of matter inside the star. The sensitivity of core $g$-mode
oscillations to the presence of a mixed phase may alleviate this difficulty. In
hybrid stars that admit quark matter in a mixed phase, the $g$-mode frequency
rises sharply due to a marked decrease in the equilibrium sound speed. Resonant
excitation of $g$-modes can leave an imprint on the waveform of coalescing
binary compact stars. We present some analytic and numeric results to assess
the sensitivity displayed by $g$-mode oscillations to quark matter in a
homogeneous or mixed phase and find that the $g$-mode can probe the fraction of
quark matter inside the star.
