  While the standard cosmological model, $\Lambda$CDM, agrees well with a host
of cosmological data on large scales, a number of contentious small-scale
tensions and the unknown physical (e.g. particle) nature of dark matter
motivate the exploration of alternative scenarios. An ultra-light bosonic
particle of mass in the range $\approx 10^{-22}\,\mathrm{eV}/c^2$ is of special
interest, as it both has particle physics motivations, and may give rise to
notable differences in the structures on highly non-linear scales due to the
manifestation of quantum-physical wave effects on macroscopic scales. Using a
spectral technique, we here discuss simulations of such fuzzy dark matter (FDM)
with a comparatively large dynamic range and for larger box sizes than
considered previously. We focus on the evolution of the matter power spectrum,
and give first results for the FDM halo mass function directly based on FDM
simulations. Unlike in $\Lambda$CDM, it is computationally challenging to
accurately treat the growth on large scales in FDM, even in the regime where
this follows linear theory. Once sufficient numerical resolution is available,
FDM shows a pronounced suppression of power on small scales relative to cold
dark matter (CDM), which can be understood as a damping effect due to 'quantum
pressure'. In certain regimes, however, the FDM power can exceed that of CDM,
which may be interpreted as a reflection of order-unity density fluctuations
occurring in FDM. The halo mass functions are identical for massive haloes, but
FDM shows a significant abundance reduction below a characteristic mass scale.
This could in principle alleviate the need to invoke very strong feedback
processes in small galaxies to reconcile $\Lambda$CDM with the observed galaxy
luminosity function, but detailed studies that also include baryons will be
needed to ultimately judge the viability of FDM.
