  Fractal lattices are self-similar structures with repeated patterns on
different scales. As in other aperiodic lattices, the absence of translational
symmetry can give rise to quantum localization effects. In contrast to
low-dimensional disordered systems, co-existence of localized and extended
states is possible in fractal structures, and can lead to subtle transport
behavior. Here, we study the dynamical properties of two fractal lattices, the
Sierpi\'nski gasket and the Sierpi\'nski carpet. Despite their geometric
similarity, the transport turns out to behave very differently: In the
Sierpi\'nski gasket, we find a sub-diffusive behavior, whereas the Sierpi\'nski
carpet exhibits sub-ballistic transport properties. We show that the different
dynamical behavior is in line with qualitative differences of the systems'
spectral properties. Specifically, in contrast to the Sierpi\'nski carpet, the
Sierpi\'nski gasket exhibits an inverse power-law behavior of the level spacing
distribution. From the point of view of technological applications, we
demonstrate that the sub-diffusive behavior in the Sierpi\'nski gasket can be
used as a quantum memory. By interpolating between fractal and regular
lattices, a flexible tuning between different transport regimes becomes
possible.
