  A nonlocal subspace $\mathcal{H}_{NS}$ is a subspace within the Hilbert space
$\mathcal{H}_n$ of a multi-particle system such that every state $\psi \in
\mathcal{H}_{NS}$ violates a given Bell inequality $\mathcal{B}$. Subspace
$\mathcal{H}_{NS}$ is maximally nonlocal if each such state $\psi$ violates
$\mathcal{B}$ to its algebraic maximum. We propose ways by which states with a
stabilizer structure of graph states can be used to construct maximally
nonlocal subspaces, essentially as a degenerate eigenspace of Bell operators
derived from the stabilizer generators. Two cryptographic applications-- to
quantum information splitting and quantum subspace certification-- are
discussed.
