We investigate the thermodynamic behavior of quantum many-body systems using several methods based on classical calculations. These approaches are compared for the melting of Lennard-Jones (LJ) clusters, where path-integral Monte Carlo (PIMC) results are also available. First, we examine two quasiclassical approaches where the classical potential is replaced by effective potentials accounting for quantum corrections of low order in h. Of the Wigner-Kirkwood and Feynman-Hibbs effective potentials, only the latter is found to be in quantitative agreement with quantum simulations. However, both potentials fail to describe even qualitatively the low-temperature regime, where quantum effects are strong. Our second approach is based on the harmonic superposition approximation, but with explicit quantum oscillators. In its basic form, this approach is in good qualitative agreement with PIMC results, and becomes more accurate at low temperatures. By including anharmonic corrections in the form of temperature-dependent frequency shifts, the agreement between the quantum superposition and the PIMC results becomes quantitative for the caloric curve of neon clusters. The superposition method is then applied to larger clusters to study the influence of quantum delocalization on the melting and premelting of LJ(19), LJ(31), LJ(38), and LJ(55). The quantum character strongly affects the thermodynamics via changes in the ground state structure due to increasing zero-point energies. Finally, we focus on the lowest temperature range, and we estimate the Debye temperatures of argon clusters and their size variation. A strong sensitivity to the cluster structure is found, especially when many surface atoms reorganize as in the anti-Mackay/Mackay transition. In the large size regime, the Debye temperature smoothly rises to its bulk limit, but still depends slightly on the growth sequence considered.