  Quantum non-locality tests have been of interest since the 1960's paper by
Bell on the original EPR paradox.The present paper discusses whether the CGLMP
(Bell) inequalities are possible tests for showing that quantum theory is not
underpinned by local hidden variable theory (LHVT). It is found by applying
Fine's theorem that the CGLMP approach involves a LHVT for the joint
probabilities associated with the measurement of one observable from each of
the two sub-systems, even though the underlying probabilities for joint
measurements of all four observables may involve a non-local HVT. The latter
HVT probabilities involve simultaneous measurements of observables
corresponding to non-commuting quantum operators - allowable in classical
theory. Although the CGLMP inequalities involve probabilities for measurements
of one observable per sub-system and are compatible with the Heisenberg
uncertainty principle, there is no unambiguous quantum measurement process
linked to the probabilities in the CGLMP inequalities, and these quantum
measurements each have different probabilities. However, violation of a CGLMP
inequality based on any one of the possible quantum measurement sequences is
sufficient to show that the Collins et al LHVT predicts different results to
quantum theory. This occurs for a state considered in their paper - though for
observables whose physical interpretation is unclear. In spite of this issue
and in spite of the contextuality loophole, it is concluded that the CGLMP
inequalities are indeed suitable for ruling out both local (and non-local)
hidden variable theories. The state involved could apply to a macroscopic
system, so the CGLMP Bell inequalities are important for finding cases of
macroscopic violations of Bell locality. Possible experiments in double-well
Bose condensates for atoms with two hyperfine components are discussed.
