  We present stellar rotation curves and velocity dispersion profiles for 104
quiescent galaxies at $z=0.6-1$ from the Large Early Galaxy Astrophysics Census
(LEGA-C) spectroscopic survey. Rotation is typically probed across 10-20kpc, or
to an average of 2.7${\rm R_e}$. Combined with central stellar velocity
dispersions ($\sigma_0$) this provides the first determination of the dynamical
state of a sample selected by a lack of star formation activity at large
lookback time. The most massive galaxies
($M_{\star}>2\times10^{11}\,M_{\odot}$) generally show no or little rotation
measured at 5kpc ($|V_5|/\sigma_0<0.2$ in 8 of 10 cases), while ${\sim}64\%$ of
less massive galaxies show significant rotation. This is reminiscent of local
fast- and slow-rotating ellipticals and implies that low- and high-redshift
quiescent galaxies have qualitatively similar dynamical structures. We compare
$|V_5|/\sigma_0$ distributions at $z\sim0.8$ and the present day by re-binning
and smoothing the kinematic maps of 91 low-redshift quiescent galaxies from the
CALIFA survey and find evidence for a decrease in rotational support since
$z\sim1$. This result is especially strong when galaxies are compared at fixed
velocity dispersion; if velocity dispersion does not evolve for individual
galaxies then the rotational velocity at 5kpc was an average of ${94\pm22\%}$
higher in $z\sim0.8$ quiescent galaxies than today. Considering that the number
of quiescent galaxies grows with time and that new additions to the population
descend from rotationally-supported star-forming galaxies, our results imply
that quiescent galaxies must lose angular momentum between $z\sim1$ and the
present, presumably through dissipationless merging, and/or that the mechanism
that transforms star-forming galaxies also reduces their rotational support.
