  If $z > 6$ quasars reside in rare, massive haloes, $\Lambda$CDM cosmology
predicts they should be surrounded by an anomalously high number of bright
companion galaxies. Here I show that these companion galaxies should also move
unusually fast. Using a new suite of cosmological, `zoom-in' hydrodynamic
simulations, I present predictions for the velocity distribution of quasar
companion galaxies and its variation with quasar host halo mass at $z \, = \,
6$. Satellites accelerate as they approach the quasar host galaxy, producing a
line-of-sight velocity profile that broadens with decreasing distance to the
quasar host galaxy. This increase in velocity dispersion is particularly
pronounced if the host halo mass is $\gtrsim 5 \times 10^{12} \, \rm M_\odot$.
Typical line-of-sight speeds rise to $\approx 500 \, \rm km \, s^{-1}$ at
projected radii $\sim 10 \, \rm kpc$. For about $10\%$ of satellites, they
should exceed $800 \, \rm km \, s^{-1}$, with $\approx 5\%$ of companions
reaching line-of-sight speeds $\sim 1000 \, \rm km \, s^{-1}$. For lower host
halo masses $\approx 5 \times 10^{11} - 10^{12} \, \rm M_\odot$, the velocity
profile of companion galaxies is significantly flatter. In this case, typical
line-of-sight velocities are $\approx 250 \, \rm km \, s^{-1}$ and do not
exceed $\approx 500 \, \rm km \, s^{-1}$. A comparison with existing ALMA, JWST
and MUSE line-of-sight velocity measurements reveals that observed $z > 6$
quasar companions closely follow the velocity distribution expected for a host
halo with mass $\gtrsim 5 \times 10^{12} \, \rm M_\odot$, ruling out a light
host halo. Finally, through an estimate of UV and \oiii \, luminosity
functions, I show that the velocity distribution more reliably discriminates
between halo mass than companion number counts, which are strongly affected by
cosmic variance.
