  The spin distribution of massive black holes (MBHs) contains rich information
on the MBH growth history. In this paper, we investigate the spin evolution of
MBHs by assuming that each MBH experiences two-phase accretion, with an initial
phase of coherent-accretion via either the standard thin disc or
super-Eddington disc, followed by a chaotic-accretion phase composed of many
episodes with different disc orientations. If the chaotic-phase is significant
to the growth of an MBH, the MBH spin quickly reaches the maximum value because
of the initial coherent-accretion, then changes to a quasi-equilibrium state
and fluctuates around a value mainly determined by the mean ratio of the disc
to the MBH mass ($M_{\bullet}$) in the chaotic-accretion episodes, and further
declines due to late chaotic-accretion if $M_\bullet \gtrsim (1-3) \times 10^8
M_\odot$. The turning point to this decline is determined by the equality of
the disc warp radius and disc size. By matching the currently available spin
measurements with mock samples generated from the two-phase model(s) on the
spin-mass plane, we find that MBHs must experience significant
chaotic-accretion phase with many episodes and the mass accreted in each
episode is roughly 1-2 percent of M_bh or less. MBHs with $M_{\bullet}\gtrsim
10^8 M_{\odot}$ appear to have intermediate-to-high spins ($\sim 0.5-1$), while
lighter MBHs have higher spins ($\gtrsim 0.8$). The best matches also infer
that (1) the radiative efficiencies ($\eta$) of those active MBHs appear to
slightly decrease with $M_{\bullet}$; however, the correlation between $\eta$
and $M_{\bullet}$, if any, is weak; (2) the mean radiative efficiency of active
MBHs is $<\eta> \sim 0.09-0.15$, consistent with the global constraints.
