  Chained correlation inequalities involving pairwise correlations of qubit
observables in the equatorial plane are constructed based on the positivity of
a sequence of moment matrices. When a jointly measurable set of fuzzy POVMs is
employed in first measurement of every pair of sequential measurements, the
chained pairwise correlations do not violate the classical bound imposed by the
moment matrix positivity. We identify that incompatibility of measurements is
only necessary, but not sufficient, in general, for the violation of the
inequality. On the other hand, there exists a one-to-one equivalence between
the degree of incompatibility (which quantifies the joint measurability) of the
equatorial qubit observables and the optimal violation of a non-local steering
inequality, proposed by Jones and Wiseman (Phys. Rev. A, 84, 012110 (2011)). To
this end, we construct a local analogue of this steering inequality in a single
qubit system and show that its violation is a mere reflection of measurement
incompatibility of equatorial qubit POVMs, employed in first measurements in
the sequential unsharp-sharp scheme.
