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Abstract 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 PQ and the true pressure P is of order P PQ − P = O $$ \mathcal{O} $$ ( α s 3 $$ {\alpha}_{\textrm{s}}^3 $$ P). The corrections arise from a single Feynman diagram; its calculation would render remaining corrections O $$ \mathcal{O} $$ ( α s 4 $$ {\alpha}_{\textrm{s}}^4 $$ P).