  The quantum Ising model with random couplings and random transverse fields on
the Cayley tree is studied by Real-Space-Renormalization in order to construct
the whole set of eigenstates. The renormalization rules are analyzed via large
deviations. The phase transition between the paramagnetic and the spin-glass
Many-Body-Localized phases involves the activated exponent $\psi=1$ and the
correlation length exponent $\nu=1$. The spin-glass-ordered cluster containing
$N_{SG}$ spins is found to be extremely sparse with respect to the total number
$N$ of spins : its size grows only logarithmically at the critical point
$N_{SG}^{criti} \propto \ln N$, and it is sub-extensive $N_{SG} \propto
N^{\theta}$ in the finite region of the spin-glass phase where the continuously
varying exponent $\theta$ remains in the interval $0<\theta<1$.
