  Clarifying similarities and differences in physical properties between
crystalline and quasicrystalline systems is one of central issues in studying
quasicrystals. To contribute to this, we apply multifractal and hyperuniform
analyses to nonuniform spatial patterns in the Bose-Hubbard model on the
Penrose and Ammann-Beenker tilings. Based on the mean-field approximation, we
obtain real-space distributions of boson density and bosonic condensate. In
both Mott insulating and superfluid phases, the distributions are hyperuniform.
Analyzing the order metric that quantifies the complexity of nonuniform spatial
patterns, we find that both quasicrystals show a divergence of the order metric
at a phase boundary between the Mott insulating and superfluid phases, in stark
contrast to the case of a periodic square lattice. Our results suggest that
hyperuniformity is a useful concept to differentiate between crystalline and
quasicrystalline bosonic systems. Moreover, we introduce on-site random
potentials into these quasicrystalline Bose-Hubbard models, leading to a Bose
glass phase. Contrary to the Mott insulating and superfluid phases, we find
that the Bose glass phase is multifractal. The same multifractality appears on
a Bose glass phase in the periodic square lattice. Therefore, multifractality
is common in a Bose glass phase irrespective of the periodicity of systems.
