  We study quantum antiferromagnetism on the highly frustrated checkerboard
lattice, also known as the square lattice with crossings. The quantum
Heisenberg antiferromagnet on this lattice is of interest as a two-dimensional
analog of the pyrochlore lattice magnet. By combining several approaches we
conclude that this system is most likely ordered for all values of spin, $S$,
with a Neel state for large $S$ giving way to a two-fold degenerate
valence-bond solid for smaller $S$. We show next that the Ising antiferromagnet
with a weak four-spin exchange, equivalent to square ice with the leading
quantum dynamics, exhibits long range ``anti-ferroelectric'' order. As a
byproduct of this analysis we obtain, in the system of weakly coupled ice
planes, a sliding phase with XY symmetry.
