  The Large Magellanic Cloud (LMC) will induce a dynamical friction (DF) wake
on infall to the Milky Way (MW). The MW's stellar halo will respond to the
gravity of the LMC and the dark matter (DM) wake, forming a stellar counterpart
to the DM wake. This provides a novel opportunity to constrain the properties
of the DM particle. We present a suite of high-resolution, windtunnel-style
simulations of the LMC's DF wake that compare the structure, kinematics, and
stellar tracer response of the DM wake in cold DM (CDM), with and without
self-gravity, vs. fuzzy DM (FDM) with $m_a = 10^{-23}$ eV. We conclude that the
self-gravity of the DM wake cannot be ignored. Its inclusion raises the wake's
density by $\sim 10\%$, and holds the wake together over larger distances
($\sim$ 50 kpc) than if self-gravity is ignored. The DM wake's mass is
comparable to the LMC's infall mass, meaning the DM wake is a significant
perturber to the dynamics of MW halo tracers. An FDM wake is more granular in
structure and is $\sim 20\%$ dynamically colder than a CDM wake, but with
comparable density. The granularity of an FDM wake increases the stars'
kinematic response at the percent level compared to CDM, providing a possible
avenue of distinguishing a CDM vs. FDM wake. This underscores the need for
kinematic measurements of stars in the stellar halo at distances of 70-100 kpc.
