Abstract: We concern on spatially homogeneous isotropic quantum Boltzmann equation for bosons with quantum collision kernel, focus on the condensation phenomenon, long time convergence to equilibrium(and its convergence rate), regularity and stability of solution. We prove that for hard potentials, the weak solution of Boltzmann equation for bosons will strong convergence to equilibrium, thus it has finite time condensation at low temperature. For balanced potentials, the weak solution of Boltzmann equation for bosons has no finite time spontaneous condensation. But it has propagation of negative order of moments , and we can use this property to get regularity and stability of the solution. 

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