  We analyze a quantum measurement $\hat{x}$ by solving the dynamics of
stochastic amplitudes that propagate both forward and backward in the time
direction. The dynamics is derived from quantum mechanics: The instantaneous
joint density of amplitudes $x$ and $p$ is proved equivalent to the positive
$Q(x,p,t)$ phase-space distribution, which establishes causal consistency. We
model the measurement $\hat{x}$ as amplification, confirming Born's rule for a
system prepared in a superposition of eigenstates $|x_{j}\rangle$ of $\hat{x}$.
The trajectories for the backward-propagating variable $x$ are governed by a
future boundary condition determined by the measurement setting, as though the
system were prepared in a mixture of $|x_{j}\rangle$. Causal relations are
deduced from the simulations. For superpositions and entangled states, we
identify causal loops for variables that are not observable. A hybrid causal
structure exists that is consistent with macroscopic realism. Further, the
model allows forward-backward simulation of Einstein-Podolsky-Rosen and Bell
correlations, which addresses a question raised by Schr\"odinger. The
simulations reveal consistency with a weak form of local realism defined for
the system after the unitary interactions determining the measurement settings,
the Bell violations emerging due to a breakdown of a subset of Bell's
local-realism conditions. Our results elucidate how hidden causal loops can
explain Bell nonlocality, without requiring retrocausality at a macroscopic
level.
