  We introduce a hierarchy of linear systems for showing that a given subspace
of pure quantum states is entangled (i.e., contains no product states). This
hierarchy outperforms known methods already at the first level, and it is
complete in the sense that every entangled subspace is shown to be so at some
finite level of the hierarchy. It generalizes straightforwardly to the case of
higher Schmidt rank, as well as the multipartite cases of completely and
genuinely entangled subspaces. These hierarchies work extremely well in
practice even in very large quantum systems, as they can be implemented via
elementary linear algebra techniques rather than the semidefinite programming
techniques that are required by previously-known hierarchies.
