  A generic semiclassical superconducting nanostructure connected to multiple
superconducting terminals hosts a quasi-continuous spectrum of Andreev states.
The spectrum is sensitive to the superconducting phases of the terminals. It
can be either gapped or gapless depending on the point in the multi-dimensional
parametric space of these phases. Special points in this space correspond to
setting some terminals to the phase 0 and the rest to the phase of $\pi$. For a
generic nanostructure, three distinct spectra come together in the vicinity of
a special point: two gapped phases of different topology and a gapless phase
separating the two by virtue of topological protection.
  In this paper, we show that a weak interaction manifesting as quantum
fluctuations of superconducting phases drastically changes the spectrum in a
narrow vicinity of a special point. We develop an interaction model and derive
a universal generic quantum action that describes this situation. The action is
complicated by incorporating a non-local in time matrix order parameter, and
its full analysis is beyond the scope of the present paper. Here, we identify
and address two limits: the semiclassical one and the quantum one,
concentrating on the first-order interaction correction in the last case.
  In both cases, we find that the interaction squeezes the domain of the
gapless phase in the narrow vicinity of the point so the gapped phases tend to
contact each other immediately, defying the topological protection. We identify
the domains of strong coupling where the perturbation theory does not work. In
the gapless phase, we find the logarithmic divergence of the first-order
corrections. This leads us to an interesting hypothesis: weak interaction might
induce an exponentially small gap in the formerly gapless phase.
