We report a theoretical result concerning the dynamics of an initially localized wave packet in quantum nonlinear Schrödinger lattices with a disordered potential. A class of nonlinear lattices with subquadratic power nonlinearity is considered. We show that there exists a parameter range for which an initially localized wave packet can spread along the lattice to unlimited distances, but the phenomenon is purely quantum and is hindered in the corresponding classical lattices. The mechanism for this spreading is moreover very peculiar and assumes that the components of the wave field may form coupled states by tunneling under the topological barriers caused by multiple discontinuities in the operator space. Then these coupled states thought of as quasiparticle states can propagate to long distances on Lévy flights with a distribution of waiting times. The overall process is subdiffusive and occurs as a competition between long-distance jumps of the quasiparticle states, on the one hand, and long-time trapping phenomena mediated by clustering of unstable modes in wave number space, on the other hand. The kinetic description of the transport, discussed in this work, is based on fractional-derivative equations allowing for both (i) non-Markovianity of the spreading process as a result of attractive interaction among the unstable modes; this interaction is then described in terms of the familiar Lennard-Jones potential; and (ii) the effect of long-range correlations in wave number space tending to introduce fast channels for the transport, the so-called "stripes." We argue that the notion of stripes is key to understand the topological constraints behind the quantum spreading, and we involve the idea of stripy ordering to obtain self-consistently the parameters of the associated waiting-time and jump-length distributions. Finally, we predict the asymptotic laws for quantum transport and show that the relevant parameter determining these laws is the exponent of the power law defining the type of the nonlinearity. The results presented here shed light on the origin of Lévy flights in quantum nonlinear lattices with disorder.