  Inspired by Lefschetz thimble theory, we treat Quantum Field Theory as a
statistical theory with a complex Probability Distribution Function (PDF). Such
complex-valued PDFs permit the violation of Bell-type inequalities, which
cannot be violated by a real-valued, non-negative PDF. In this paper, we
consider the Classical-Statistical approximation in the context of Bell-type
inequalities, viz. the familiar (spatial) Bell inequalities and the temporal
Leggett-Garg inequalities. We show that the Classical-Statistical approximation
does not violate temporal Bell-type inequalities, even though it is in some
sense exact for a free theory, whereas the full quantum theory does. We explain
the origin of this discrepancy, and point out the key difference between the
spatial and temporal Bell-type inequalities. We comment on the import of this
work for applications of the Classical-Statistical approximation.
