  Quantum probes using $N$ uncorrelated particles give a limit on the
measurement sensitivity referred to as the standard quantum limit (SQL). The
SQL, however, can be overcome by exploiting quantum entangled states, such as
spin squeezed states. We report generation of a quantum state, that surpasses
the SQL for probing of the collective spin of $10^{11}$ $\text{Rb}$ atoms
contained in a vapor cell. The state is prepared and verified by sequences of
stroboscopic quantum non-demolition (QND) measurements, and we apply the theory
of past quantum states to obtain the spin state information from the outcomes
of both earlier and later QND measurements. In this way, we obtain a
conditional noise reduction of 5.6 dB, and a metrologically-relevant squeezing
of $4.5\pm0.40~\text{dB}$. The past quantum state yields tighter information on
the spin component than we can obtain by a conventional QND measurement. Our
squeezing results are obtained with 1000 times more atoms than in any previous
experiments with a corresponding record $4.6\times10^{-13} rad^2$ variance of
the angular fluctuations of a squeezed collective spin.
