The equilibrium lattice constants, elastic constants and elastic moduli of wII- and post-spinel Si₃N₄ have been investigated in the pressure ranges of 0-40 GPa and 160-240 GPa, respectively. These two phases are found to be dynamically stable. The post-spinel phase is one of the strongest materials yet investigated under extreme compressions. Some fundamental properties and phase transition characters are evaluated from the quasi-harmonic approximation. The transition pressures from the γ phase to the post-spinel phase are 152.5 GPa (at 300 K) and 181.8 GPa (at 1500 K). The phase transition pressures of the β→wII andγ→post-spinel transitions increase with the rise of temperature; hence, at higher temperature it requires higher pressure to synthesize post-spinel Si₃N₄. The heat capacity, thermal expansion and bulk modulus of the new phases are computed as functions of pressure and temperature. The heat capacity of post-spinel Si₃N₄ is large at high temperature and only weakly pressure dependent.