  A key property of Majorana zero modes is their protection against local
perturbations. In the standard picture, this protection is guaranteed by a high
degree of spatial nonlocality of the Majoranas, namely a suppressed
wave-function overlap, in the topological phase. However, a careful
characterisation of resilience to local noise goes beyond mere spatial
separation, and must also take into account the projection of wave-function
spin. By considering the susceptibility of a given zero mode to different local
perturbations, we find the relevant forms of spin-resolved wave-function
overlaps that measure its resilience. We quantify these overlaps and study
their dependence with nanowire parameters in several classes of experimentally
relevant configurations. These include nanowires with inhomogeneous depletion
and induced pairing, barriers and quantum dots. Smooth inhomogeneities have
been shown to produce near-zero modes, so-called pseudo-Majoranas, below the
critical Zeeman field in the bulk. Surprisingly, their resilience is found to
be comparable or better than that of topological Majoranas in realistic
systems. We further study how accurately their overlaps can be estimated using
a purely local measurement on one end of the nanowire, accessible through
conventional transport experiments. In uniform nanowires this local estimator
is remarkably accurate. In inhomogeneous cases it is less accurate but can
still provide reasonable estimates for potential inhomogeneities of the order
of the superconducting gap. We further analyse the zero mode wave-function
structure, spin texture and spectral features associated with each type of
inhomogeneity. All our results highlight the strong connection between internal
wave-function degrees of freedom, nonlocality and protection in smoothly
inhomogeneous nanowires.

Mesoscale and Nanoscale Physics