  A Bloch sphere is the geometrical representation of an arbitrary
two-dimensional Hilbert space. Possible classes of entanglement and
separability for the pure and mixed states on the Bloch sphere were suggested
by [M. Boyer, R. Liss, T. Mor, PRA 95, 032308 (2017)]. Here we construct a
Bloch sphere for the Hilbert space spanned by one of the ground states of
Kitaev's toric code model and one of its closest product states. We prove that
this sphere contains only one separable state, thus belonging to the fourth
class suggested by the said paper. We furthermore study the topological order
of the pure states on its surface and conclude that, according to conventional
definitions, only one state (the toric code ground state) seems to present
non-trivial topological order. We conjecture that most of the states on this
Bloch sphere are neither ``trivial'' states (namely, they cannot be generated
from a product state using a trivial circuit) nor topologically ordered. In
addition, we show that the whole setting can be understood in terms of Grover
rotations with gauge symmetry, akin to the quantum search algorithm.
