We examine the possibility of quasicrystalline metastable states for one- and two-component hard-sphere solids, systems whose excess free energies are entirely entropic. The investigations are carried out using the self-consistent modified weighted density approximation of density functional theory as applied to inhomogeneous systems. By a method entirely formulated in reciprocal space, we find that the best quasicrystalline phases have significantly higher free energies than the crystalline or liquid states. These results indicate that quasicrystals probably cannot be stabilized entirely by dynamic entropy.