  Magnetic critical current density and relaxation rate have been measured on
$MgB_2$ bulks from 1.6 K to $T_c$ at magnetic fields up to 8 Tesla. A vortex
phase diagram is depicted based on these measurement. Two phase boundaries
$H_{irr}^{bulk}(T)$ and $H_{irr}^{g}(T)$ characterizing different irreversible
flux motions are found. The $H_{irr}^{bulk}(T)$ is characterized by the
appearance of the linear resistivity and is attributed to quantum vortex
melting induced by quantum fluctuation of vortices in the rather clean system.
The second boundary $ H_{irr}^g(T) $ reflects the irreversible flux motion in
some local regions due to either very strong pinning or the surface barrier on
the tiny grains.

Superconductivity