  The internal composition of neutron stars is currently largely unknown. Due
to the possibility of phase transitions in quantum chromodynamics, stars could
be hybrid and have quark cores. We investigate some imprints of elastic quark
phases (only when perturbed) on the dynamical stability of hybrid stars. We
show that they increase the dynamical stability window of hybrid stars in the
sense that the onset of instabilities happen at larger central densities than
the ones for maximum masses. In particular, when the shear modulus of a
crystalline quark phase is taken at face value, the relative radius differences
between elastic and perfect-fluid hybrid stars with null radial frequencies
(onset of instability) would be up to $1-2\%$. Roughly, this would imply a
maximum relative radius dispersion (on top of the perfect-fluid predictions) of
$2-4\%$ for stars in a given mass range exclusively due to the elasticity of
the quark phase. In the more agnostic approach where the estimates for the
quark shear modulus only suggest its possible order of magnitude (due to the
many approximations taken in its calculation), the relative radius dispersion
uniquely due to a quark phase elasticity might be as large as $5-10\%$.
Finally, we discuss possible implications of the above dispersion of radii for
the constraint of the elasticity of a quark phase with electromagnetic missions
such as NICER, eXTP and ATHENA.
