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The quantum Ising model with random couplings and random transverse fields onthe Cayley tree is studied by Real-Space-Renormalization in order to constructthe whole set of eigenstates. The renormalization rules are analyzed via largedeviations. The phase transition between the paramagnetic and the spin-glassMany-Body-Localized phases involves the activated exponent $\psi=1$ and thecorrelation 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 $N_{SG} \proptoN^{\theta}$ in the finite region of the spin-glass phase where the continuouslyvarying exponent $\theta$ remains in the interval $0<\theta<1$.

Random Transverse Field Spin-Glass Model on the Cayley tree : phase transition between the two Many-Body-Localized Phases