Several recent experiments in atomic, molecular and optical systems motivated a huge interest in the study of quantum long-range spin systems. Our goal is to discuss and present a general description of their critical behavior and phases, devising a treatment valid in d dimensions, with an exponent d+\sigma for the power-law decay of the couplings in the presence of an O(N) symmetry. By introducing a convenient ansatz for the effective action, we determine the phase diagram for the N-component quantum rotor model with long-range interactions, with N=1 corresponding to the Ising model. The phase diagram in the \sigma−d plane shows a non trivial dependence on \sigma. As a consequence of the fact that the model is quantum, the correlation functions are anisotropic in the spatial and time coordinates for σ smaller than a critical value and in this region the isotropy is not restored even at criticality. Results for the correlation length exponent, the dynamical critical exponent z and a comparison with numerical findings for them are presented.