  The Schr\"odinger-Poisson equations describe the behavior of a superfluid
Bose-Einstein condensate under self-gravity with a 3D wave function. As
$\hbar/m\to 0$, $m$ being the boson mass, the equations have been postulated to
approximate the collisionless Vlasov-Poisson equations also known as the
collisionless Boltzmann-Poisson equations. The latter describe collisionless
matter with a 6D classical distribution function. We investigate the nature of
this correspondence with a suite of numerical test problems in 1D, 2D, and 3D
along with analytic treatments when possible. We demonstrate that, while the
density field of the superfluid always shows order unity oscillations as
$\hbar/m\to 0$ due to interference and the uncertainty principle, the potential
field converges to the classical answer as $(\hbar/m)^{2}$. Thus, any dynamics
coupled to the superfluid potential is expected to recover the classical
collisionless limit as $\hbar/m\to 0$. The quantum superfluid is able to
capture rich phenomena such as multiple phase-sheets, shell-crossings, and warm
distributions. Additionally, the quantum pressure tensor acts as a regularizer
of caustics and singularities in classical solutions. This suggests the
exciting prospect of using the Schr\"odinger-Poisson equations as a low-memory
method for approximating the high-dimensional evolution of the Vlasov-Poisson
equations. As a particular example we consider dark matter composed of
ultra-light axions, which in the classical limit ($\hbar/m\to 0$) is expected
to manifest itself as collisionless cold dark matter.
