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 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 \times 10 ^ { 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 \sim 0.8 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 \sim 1 . 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 \pm 22 \% } higher in z \sim 0.8 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 \sim 1 and the present , presumably through dissipationless merging , and/or that the mechanism that transforms star-forming galaxies also reduces their rotational support .