  We explore the impact of long-range memory on the properties of a family of
quantum walks in a one-dimensional lattice and discrete time, which can be
understood as the quantum version of the classical "Elephant Random Walk"
non-Markovian process. This Elephant Quantum Walk is robustly superballistic
with the standard deviation showing a constant exponent, $\sigma \propto t^{3/
2}$ , whatever the quantum coin operator, on which the diffusion coefficient is
dependent. On the one hand, this result indicates that contrarily to the
classical case, the degree of superdiffusivity in quantum non- Markovian
processes of this kind is mainly ruled by the extension of memory rather than
other microscopic parameters that explicitly define the process. On the other
hand, these parameters reflect on the diffusion coefficient.
