  Sun-like stars are thought to accrete most of their final mass during the
protostellar phase, during which the stellar embryo is surrounded by an
infalling dense envelope. We present an analysis of 26 $K$-band spectra of
Class 0 protostars, which are the youngest protostars. 18 of these are new
observations made with the Keck MOSFIRE instrument. HI Br$\gamma$ , several
H$_2$, and CO $\Delta\,v\;=\;2$ features are detected and analyzed. We detect
Br$\gamma$ emission in 62%, CO overtone emission in 50%, and H$_2$ emission in
90% of sources. The HI and CO emission is associated with accretion while the
H$_2$ lines are consistent with shock excitation indicating jets/outflows. Six
objects exhibit photospheric absorption features, with almost no outflow
activity, and no detection of the accretion-related Br$\gamma$ emission line.
Comparing these results with archival sample of Class I $K$-band spectra, we
find that the CO and Br$\gamma$ emission lines are systematically more luminous
in Class 0s, suggesting the accretion is on average more vigorous in the Class
0 phase. Typically associated with the heated inner accretion disk, the much
higher detection rate of CO overtone emission in Class 0s indicate also that
episodes of high accretion activity are more frequent in Class 0 systems. The
kinematics of the Class 0 CO overtone emission suggest either an
accretion-heated inner disk, or material directly infalling onto the central
region. This could point toward an accretion mechanism of different nature in
Class 0 systems than the typical picture of magnetospheric accretion.
