  In this paper we focus on the underlying quantum structure of temporal
correlations and show their peculiar nature which differentiate them from
spatial quantum correlations. We show rigorously that a particular entangled
history, which can be associated with a quantum propagator, is monogamous to
conserve its consistency throughout time. Yet evolving systems violate
monogamous Bell-like multi-time inequalities. This dichotomy, being a novel
feature of temporal correlations, has its roots in the measurement process
itself which is discussed by means of the bundles of entangled histories. We
introduce and discuss a concept of a probabilistic mixture of quantum processes
by means of which we clarify why the spatial-like Bell-type monogamous
inequalities are further violated. We prove that Tsirelson bound on temporal
Bell-like inequalities can be derived from the entangled histories approach and
as a generalization, we derive the quantum bound for multi-time Bell-like
inequalities. It is also pointed out that what mimics violation of monogamy of
temporal entanglement is actually just a kind of polyamory in time but monogamy
of entanglement for a particular evolution still holds.
