  We consider here the predictions of quantum theory and local hidden variables
for the correlations obtained by measuring a pair of qubits by projections
defined by randomly chosen axes separated by a given angle \theta. The
predictions of local hidden variable models for projective measurements on
qubits correspond to binary colourings of the Bloch sphere with antipodal
points oppositely coloured. We prove Bell inequalities separating the
predictions of all local hidden variable models from the singlet correlations
predicted by quantum theory for all \theta in the range 0 < \theta < \pi/3. We
raise and explore the possibility of proving stronger Bell inequalities
directly from optimization results on sphere colourings. In particular, we
explore strong and weak forms of the hemispherical colouring maximality
hypothesis (HCMH) that, for a continuous range of \theta > 0, the maximum LHV
anti-correlation is obtained by assigning to each qubit a colouring with one
hemisphere black and the other white. Our results show that hemispherical
colourings are near-optimal for small \theta; we also describe numerical tests
consistent with the HCMH that bound the range of \theta. Finally, we note
proofs of related results for binary colourings of R^n.
