  The Future Circular Collider (FCC) -- in its first incarnation as a lepton
collider -- will produce, according to the proposed design, more than 100
billion pairs of $\tau$ leptons after working for four years at the energy of
the $Z$-boson resonance. The $\tau$ lepton is special because its relatively
long lifetime makes it possible to reconstruct the momenta of neutrinos emitted
in the single pion decay mode. The resulting large number of events is an ideal
source for a full quantum tomography of the process that will test quantum
entanglement and the violation of Bell inequality with unprecedented precision.
In addition, the study of polarizations and spin correlations can provide a
competitive determination of the Weinberg angle $ \theta_W$ and constrain
possible anomalous couplings in the neutral electroweak current. We utilize
analytic results and Monte Carlo simulations to explore to what extent these
goals might be accomplished.
