  An orthonormal basis consisting of unentangled (pure tensor) elements in a
tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In
general, for $n$ qubits, we prove that in its natural structure as a real
variety, the space of UOB is a bouquet of products of Riemann spheres
parametrized by a class of edge colorings of hypercubes. Its irreducible
components of maximum dimension are products of $2^n-1$ two-spheres. Using a
theorem of Walgate and Hardy, we observe that the UOB whose elements are
distinguishable by local operations and classical communication (called locally
distinguishable or LOCC distinguishable UOB) are exactly those in the maximum
dimensional components. Bennett et al, in their in-depth study of quantum
nonlocality without entanglement, include a specific 3 qubit example UOB which
is not LOCC distinguishable; we construct certain generalized counterparts of
this UOB in $n$ qubits.
