We report the results of an experimental study of the silica phases (stishovite and SiO2 with alpha -PbO2 like structure). These phases were synthesized in laser- and externally electrically heated Mao-Bell type diamond anvil cells (DACs) from a-cristobalite as a starting material at pressures above 80 GPa and temperatures above 1200 K. We measured the hardness (H) of both phases and found that the value for the alpha -PbO2-like silica, H=31+/-3 GPa, is very close to that of stishovite, H=32+/-2 GPa, for polycrystalline material obtained in electrically heated DACs. Thus, these two polycrystalline silica phases are hardest known oxides and representatives of hardest known polycrystalline materials in general.