  We investigate a large number of cosmological evolution models for QSOs and
Active Galactic Nuclei (AGN). We introduce a spectrum-luminosity correlation as
a new input parameter and adopt the estimated mass function (MF) of massive
black holes in centers of nearby galactic nuclei as a constraint to distinguish
among different QSO/AGN models. We explore three basic types of
phenomenological scenarios; (i) Models with multiple short-lived ($\sim$ a few
$10^{6-8}$ yrs) populations, (ii) Models with a single long-lived ($\sim 10^9$
yr) QSO population, and (iii) Models with recurrent QSO/AGN activities which
are driven by long-term variabilities of the disk instability type. In each
model, we derive the expected theoretical luminosity function (LF) and the MF
of black holes which grow through mass accretion. We assess the plausibility of
each model based on whether each model's LF and MF are compatible with the
observed data. We find that the best fits to the observed LFs are obtained in
the model with multiple short-lived populations and without any significant
spectral evolution. This finding suggests that the QSO populations may be
composed of many short-lived generations ($\sim$ a few 10$^8$ yrs) and that
there is no significant spectral evolution within each generation. On the other
hand, we also show that there is no satisfactory model which can simultaneously
account for the observed LF and the estimated MF. We speculate that some of the
present-day black holes (BHs) found in galactic nuclei may have formed without
undergoing the QSO/AGN phase.
