Bloch-sphere colorings and Bell inequalities

We consider the quantum and local hidden variable (LHV) correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by an angle theta. Local hidden variables predict binary colorings of the Bloch sphere with antipodal points oppositely colored. We prove Bell inequalities separating the LHV predictions from the singlet quantum correlations for theta is an element of (0, pi/3). We raise and explore the hypothesis that, for a continuous range of theta > 0, the maximum LHV anticorrelation is obtained by assigning to each qubit a coloring with one hemisphere black and the other white.