  We add quantum fluctuations to a classical period-doubling Hamiltonian time
crystal, replacing the $N$ classical interacting angular momenta with quantum
spins of size $l$. The full permutation symmetry of the Hamiltonian allows a
mapping to a bosonic model and the application of exact diagonalization for
quite large system size. In the thermodynamic limit $N\to\infty$ the model is
described by a system of Gross-Pitaevskii equations whose classical-chaos
properties closely mirror the finite-$N$ quantum chaos. For $N\to\infty$, and
$l$ finite, Rabi oscillations mark the absence of persistent period doubling,
which is recovered for $l\to\infty$ with Rabi-oscillation frequency tending
exponentially to 0. For the chosen initial conditions, we can represent this
model in terms of Pauli matrices and apply the discrete truncated Wigner
approximation. For finite $l$ this approximation reproduces no Rabi
oscillations but correctly predicts the absence of period doubling. Our results
show the instability of time-translation symmetry breaking in this classical
system even to the smallest quantum fluctuations, because of tunneling effects.
