  We investigate the properties of a Fibonacci quasicrystal (QC) arrangement of
a one-dimensional topological superconductor, such as a magnetic atom chain
deposited on a superconducting surface. We uncover a general mutually exclusive
competition between the QC properties and the topological superconducting phase
with Majorana bound states (MBS): there are no MBS inside the QC gaps and the
MBS never behaves as QC subgap states, and likewise, no critical, or winding,
QC subgap states exist inside the topological superconducting gaps.
Surprisingly, despite this competition, we find that the QC is still highly
beneficial for realizing topological superconductivity with MBS. It both leads
to additional large nontrivial regions with MBS in parameter space, that are
topologically trivial in crystalline systems, and increases the topological gap
protecting the MBS. We also find that shorter approximants of the Fibonacci QC
display the largest benefits. As a consequence, our results promote QCs, and
especially their short approximants, as an appealing platform for improved
experimental possibilities to realize MBS as well as generally highlights the
fundamental interplay between different topologies.
