  The equation of state of QCD matter at high densities is relevant for neutron
star structure and for neutron star mergers and has been a focus of recent
work. We show how lattice QCD simulations, free of sign problems, can provide
an upper bound on the pressure as a function of quark chemical potentials. We
show that at large chemical potentials this bound should become quite sharp;
the difference between the upper bound on the pressure P-phase-quenched and the
true pressure P is of order alpha^3 P. The corrections arise from a single
Feynman diagram; its calculation would render remaining corrections of order
alpha^4 P.
