  A quantum walk on a toral phase space involving translations in position and
its conjugate momentum is studied in the simple context of a coined walker in
discrete time. The resultant walk, with a family of coins parametrized by an
angle is such that its spectrum is exactly solvable with eigenangles for odd
parity lattices being equally spaced, a feature that is remarkably independent
of the coin. The eigenvectors are naturally specified in terms the
$q-$Pochhammer symbol, but can also be written in terms of elementary
functions, and their entanglement can be analytically found. While the phase
space walker shares many features in common with the well-studied case of a
coined walker in discrete time and space, such as ballistic growth of the
walker position, it also presents novel features such as exact periodicity, and
formation of cat-states in phase-space. Participation ratio (PR) a measure of
delocalization in walker space is studied in the context of both kinds of
quantum walks; while the classical PR increases as $\sqrt{t}$ there is a time
interval during which the quantum walks display a power-law growth $\sim
t^{0.825}$. Studying the evolution of coherent states in phase space under the
walk enables us to identify an Ehrenfest time after which the coin-walker
entanglement saturates.
