  In this paper the compatibility is analyzed of the non-perturbative equations
of state of quarks and gluons arising from the lattice with some natural
requirements for self-gravitating objects at equilibrium: the existence of an
equation of state (namely, the possibility to define the pressure as a function
of the energy density), the absence of superluminal propagation and Le
Chatelier's principle. It is discussed under which conditions it is possible to
extract an equation of state (in the above sense) from the non-perturbative
propagators arising from the fits of the latest lattice data. In the quark
case, there is a small but non-vanishing range of temperatures in which it is
not possible to define a single-valued functional relation between density and
pressure. Interestingly enough, a small change of the parameters appearing in
the fit of the lattice quark propagator (of around 10~\%) could guarantee the
fulfillment of all the three conditions (keeping alive, at the same time, the
violation of positivity of the spectral representation, which is the expected
signal of confinement). As far as gluons are concerned, the analysis shows very
similar results. Whether or not the non-perturbative quark and gluon
propagators satisfy these conditions can have a strong impact on the estimate
of the maximal mass of quark stars.
