I will present our most recent numerical results on thermodynamics of quantum Lennard Jones (LJ) clusters. The classical LJ(n) clusters had been studied previously and very extensively, particularly in our group: J. Chem. Phys. 2006, 124, 20451; J. Phys. Chem. , 2007, 111, 10284. The above numerical results demonstrate how complex these systems can be (even without taking into account the quantum effects), having a number of structural motifs that are thermodynamically stable for different ranges of the size n and temperature T. The quantm effects change the critical parameters (n and T) as we demonstrated for the case of neon clusters: J. Chem. Phys. 2005, 122, 154305; Phys. Rev Lett. 2006, 96, 113401 Our most recent results explore the evolution of the ground state (i.e., the zero temperature case) of the LJ(n,L) cluster as a function of its size n and the de Boer quantum delocalization length L. The corresponding "phase diagram" that we constructed shows the stability ranges for the competing structural motifs. An increase of L has an effect similar to heating and as such may induce structural transformations.