  We present the redshift evolution of quasar luminosity functions decomposed
by halo mass, galaxy mass, supermassive black hole (SMBH) mass, and Eddington
ratio, as well as SMBH kinetic/radiative energy output ratios from TRINITY, a
flexible empirical model that self-consistently infers the halo--galaxy--SMBH
connection that match observational data. Key findings include: 1) The
normalization of QLF increases by ~3-4 dex from z~10 to z~4, due to the fast
mass build-up of different SMBH populations; 2) From z~4 to z~1, less massive
galaxies and SMBHs make up bigger and bigger fractions of QLFs, due to the AGN
downsizing effect; 3) At z~0, massive haloes/galaxies/SMBHs are responsible for
most bright quasars due to low Eddington ratios among all SMBHs; 4) The bright
ends of quasar luminosity functions (QLFs) are dominated by SMBHs that are at
least 0.3 dex over-massive relative to the median SMBH mass-galaxy mass
relation; 5) QLFs at z~6-7 are dominated by SMBHs accreting at Eddington ratios
0.1 < $\eta_\mathrm{rad}$ < 1, but super-Eddington AGNs contribute more
significantly to QLFs towards z~9-10.
