  The Hamiltonian Mean Field (HMF) model is a prototype for systems with
long-range interactions. It describes the motion of $N$ particles moving on a
ring, coupled through an infinite-range potential. The model has a second order
phase transition at the energy $U_c=3/4$ and its dynamics is exactly described
by the Vlasov equation in the $N \to \infty$ limit. Its chaotic properties have
been investigated in the past, but the determination of the scaling with $N$ of
the Lyapunov Spectrum (LS) of the model remains a challenging open problem. We
here show that the $N^{-1/3}$ scaling of the Maximal Lyapunov Exponent (MLE),
found in previous numerical and analytical studies, extends to the full LS; not
only, scaling is "precocious" for the LS, meaning that it becomes manifest for
a much smaller number of particles than the one needed to check the scaling for
the MLE. Besides that, the $N^{-1/3}$ scaling appears to be valid not only for
$U>U_c$, as suggested by theoretical approaches based on a random matrix
approximation, but also below a threshold energy $U_t \approx 0.2$. Using a
recently proposed method (GALI) devised to rapidly check the chaotic or regular
nature of an orbit, we find that $U_t$ is also the energy at which a sharp
transition from {\it weak} to {\it strong} chaos is present in the phase-space
of the model. Around this energy the phase of the vector order parameter of the
model becomes strongly time dependent, inducing a significant untrapping of
particles from a nonlinear resonance.
