  Coherent many-body states are highly promising for robust and scalable
quantum information processing. While far-reaching theoretical predictions have
been made for various implementations, direct experimental evidence of their
appealing properties can be challenging. Here, we demonstrate coherent optical
manipulation of the nuclear spin ensemble in the lead halide perovskite
semiconductor FAPbBr$_3$ (FA=formamidinium), targeting a long-postulated
collective dark state that is insensitive to optical pumping. Via optical
orientation of localized hole spins we drive the nuclear many-body system into
an entangled state, requiring a weak magnetic field of only a few Millitesla
strength at cryogenic temperatures. During its fast build-up, the nuclear
polarization along the optical axis remains small, while the transverse nuclear
spin fluctuations are strongly reduced, corresponding to spin squeezing as
evidenced by a strong violation of the generalized nuclear squeezing-inequality
with $\xi_s < 0.3$. The dark state evidenced in this process corresponds to an
approximately 750-body entanglement between the nuclei. Dark nuclear spin
states can be exploited to store quantum information benefiting from their
long-lived many-body coherence and to perform quantum measurements with a
precision beyond the standard limit.
