  Twin-peak quasi-periodic oscillations (QPOs) are observed in the X-ray
power-density spectra of several accreting low-mass neutron star (NS) binaries.
In our previous work we have considered several QPO models. We have identified
and explored mass-angular-momentum relations implied by individual QPO models
for the atoll source 4U 1636-53. In this paper we extend our study and confront
QPO models with various NS equations of state (EoS). We start with simplified
calculations assuming Kerr background geometry and then present results of
detailed calculations considering the influence of NS quadrupole moment
(related to rotationally induced NS oblateness) assuming Hartle-Thorne
spacetimes. We show that the application of concrete EoS together with a
particular QPO model yields a specific mass-angular-momentum relation. However,
we demonstrate that the degeneracy in mass and angular momentum can be removed
when the NS spin frequency inferred from the X-ray burst observations is
considered. We inspect a large set of EoS and discuss their compatibility with
the considered QPO models. We conclude that when the NS spin frequency in 4U
1636-53 is close to 580Hz we can exclude 51 from 90 of the considered
combinations of EoS and QPO models. We also discuss additional restrictions
that may exclude even more combinations. Namely, there are 13 EOS compatible
with the observed twin peak QPOs and the relativistic precession model.
However, when considering the low frequency QPOs and Lense-Thirring precession,
only 5 EOS are compatible with the model.
