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We propose a versatile setup that allows performing spin-resolved spectroscopy of helical Andreev bound states, proving their existence and therefore the topological nature of the Josephson junction that hosts them. The latter is realized on the helical edge of a 2D topological insulator, proximitized with two superconducting electrodes. The spectroscopic analysis is enabled by a quantum point contact, which couples the junction with another helical edge acting as a spin-sensitive probe. By means of straightforward transport measurements, it is possible to detect the particular spin structure of the helical Andreev bound state and to shine light on the mechanism responsible for their existence, i.e., the presence of protected Andreev reflection within the Josephson junction. We discuss the robustness of this helical Andreev spectrometer with respect to processes that weaken spin to charge conversion.

Majorana fermions feature non-Abelian exchange statistics and promise fascinating applications in topological quantum computation. Recently, second-order topological superconductors (SOTSs) have been proposed to host Majorana fermions as localized quasiparticles with zero excitation energy, pointing out a new avenue to facilitate topological quantum computation. We provide a minimal model for SOTSs and systematically analyze the features of Majorana zero modes with analytical and numerical methods. We further construct the fundamental fusion principles of zero modes stemming from single or multiple SOTS islands. Finally, we propose concrete schemes in different setups formed by SOTSs, enabling us to exchange and fuse the zero modes for non-Abelian braiding and holonomic quantum gate operations.

Majorana bound states are zero-energy excitations of topological superconductors that obey non-Abelian exchange statistics and are basic building blocks for topological quantum computation. To observe and exploit their extraordinary properties, we need to be able to properly manipulate them, for instance by braiding a couple of them in real space. We propose a setup based on the helical edges of two-dimensional topological insulators (2DTIs) that allows for a high degree of tunability by only controlling a handful of superconducting phases. In particular, our setup allows us to move the Majoranas along a single edge and to move them across two different edges coupled by a quantum point contact. Robustness against nonoptimal control of the phases is also discussed. This proposal constitutes an essential step forward toward realizing 2DTI-based architectures capable of performing braiding of Majoranas in a feasible way.