  We present the velocity dispersion measurements of four massive
$\sim10^{11}M_\odot$ quiescent galaxies at $3.2 < z < 3.7$ based on deep H and
K$-$band spectra using the Keck/MOSFIRE near-infrared spectrograph. We find
high velocity dispersions of order $\sigma_e\sim250$ km/s based on strong
Balmer absorption lines and combine these with size measurements based on
HST/WFC3 F160W imaging to infer dynamical masses. The velocity dispersion are
broadly consistent with the high stellar masses and small sizes. Together with
evidence for quiescent stellar populations, the spectra confirm the existence
of a population of massive galaxies that formed rapidly and quenched in the
early universe $z>4$. Investigating the evolution at constant velocity
dispersion between $z\sim3.5$ and $z\sim2$, we find a large increase in
effective radius $0.35\pm0.12$ dex and in dynamical-to-stellar mass ratio
$<$log(M$_{\textrm{dyn}}$/M*)$>$ of 0.33$\pm0.08$ dex, with low expected
contribution from dark matter. The dynamical masses for our $z\sim3.5$ sample
are consistent with the stellar masses for a Chabrier initial mass function
(IMF), with the ratio $<$log(M$_{\textrm{dyn}}$/M$^*_{\textrm{Ch}})>$ =
-0.13$\pm$0.10 dex suggesting an IMF lighter than Salpeter may be common for
massive quiescent galaxies at $z>3$. This is surprising in light of the
Salpeter or heavier IMFs found for high velocity dispersion galaxies at
$z\sim2$ and cores of present-day ellipticals, which these galaxies are thought
to evolve into. Future imaging and spectroscopic observations with resolved
kinematics using the upcoming James Webb Space Telescope could rule out
potential systematics from rotation, and confirm these results.
