  Both fast and slow configurations of rotating neutron stars (NSs) are studied
with the recently-constructed unified NS EoSs. The calculations for pure quark
stars (QSs) and hybrid stars (HSs) are also done, using several updated quark
matter EoSs and Gibbs construction for obtaining hadron-quark mixed phase. All
three types of EoSs fulfill the recent 2-solar-mass constrain. By confronting
the glitch observations with the theoretical calculations for the crustal
moment of inertia (MoI), we find that the glitch crisis is still present in
Vela-like pulsars. An upcoming accurate MoI measurement (eg., a possible 10\%
accuracy for pulsar PSR J0737-3039A) could distinguish QSs from NSs, since the
MoIs of QSs are generally $> \sim 1.5 $ times larger than NSs and HSs, no
matter the compactness and the mass of the stars. Using tabulated EoSs, we
compute stationary and equilibrium sequences of rapidly rotating, relativistic
stars in general relativity from the well-tested $rns$ code, assuming the
matter comprising the star to be a perfect fluid. All three observed properties
of the short gamma-ray bursts (SGRBs) internal plateaus sample are simulated
using the rotating configurations of NSs/QSs as basic input. We finally argue
that for some characteristic SGRBs, the post-merger products of NS-NS mergers
are probably supramassive QSs rather than NSs, and NS-NS mergers are a
plausible location for quark deconfinement and the formation of QSs.
