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 ($M_{star}>2times10^{11},M_{odot}$) 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 $zsim0.8$ 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 $zsim1$. 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 ${94pm22%}$ higher in $zsim0.8$ 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 $zsim1$ and the present, presumably through dissipationless merging, and/or that the mechanism that transforms star-forming galaxies also reduces their rotational support.
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