We study observational constraints on the cosmic evolution of the relationships between massive black hole (MBH) mass (M • ) and stellar mass (M * ,sph ; or velocity dispersion σ ) of a host galaxy/spheroid. Assuming that the M • -M * ,sph (or M • -σ ) relation evolves with redshift as ∝ (1 + z) Γ , the MBH mass density can be obtained from either the observationally determined galaxy stellar mass functions or velocity dispersion distribution functions over redshift z ∼ 0-1.2 for any given Γ. The MBH mass density at different redshifts can also be inferred from the luminosity function of QSOs/active galactic nuclei (AGNs) provided the radiative efficiency is known. By matching the MBH density inferred from galaxies to that obtained from QSOs/AGNs, we find that Γ = 0.64 +0.27 −0.29 for the M • -M * ,sph relation and Γ = −0.21 +0.28 −0.33 for the M • -σ relation, and = 0.11 +0.04 −0.03 . Our results suggest that MBH mass growth precedes bulge mass growth but that the galaxy velocity dispersion does not increase with the mass growth of the bulge after nuclear activity is quenched, which is roughly consistent with the two-phase galaxy formation scenario proposed by Oser et al. in which a galaxy roughly doubles its mass after z = 1 due to accretion and minor mergers while its velocity dispersion drops slightly.

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 • ) in the chaotic-accretion episodes, and further declines due to late chaoticaccretion if M • (1 − 3) × 10 8 M ⊙ . 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 • or less. MBHs with M • 10 8 M ⊙ appear to have intermediate-tohigh spins (∼ 0.5 − 1), while lighter MBHs have higher spins ( 0.8). The best matches also infer that (1) the radiative efficiencies (η) of those active MBHs appear to slightly decrease with M • ; however, the correlation between η and M • , if any, is weak; (2) the mean radiative efficiency of active MBHs is η ∼ 0.09 − 0.15, consistent with the global constraints.