Andreev bound states are coherent electron-hole superpositions that form in a normal metal through repeated Andreev reflection at a superconducting interface. When the length of a superconducting segment is comparable to the coherence length, the bound states on opposite sides of the segment hybridize through quasiparticle tunneling. In a periodic array, these hybridized Andreev bound states form energy bands below the superconducting gap. We develop a theoretical framework for transport in such Andreev crystals. We demonstrate that, at high interface transparency, a constant phase bias between neighboring superconductors renders the bands directional: one band contains only right-moving and the other only left-moving electronic states. This property leads to a directional conductance that enables the device to operate as a flux- and bias-voltage-tunable filter that allows signal transmission in only one direction.

Altermagnetic superconductors offer the possibility of exploring unconventional superconductivity, including topological states and finite-momentum superconductivity, with promising applications in spintronics and quantum information. However, a direct experimental confirmation of their existence remains elusive. In this work, we propose nonmagnetic impurities as probes of the interplay between altermagnetism and superconductivity. These impurities induce spin-polarized subgap states whose spatial extension reflects the magnetic order of the substrate material. Depending on whether or not the impurity respects the bulk symmetries, these states form spin-degenerate or spin-split doublets. An external magnetic field aligned with the Néel vector can further control the doublet splitting. We further demonstrate that coupling between impurity states leads to a position-dependent, spin-sensitive hybridization, enabling another approach for in situ control of atomic-size quantum devices. These findings provide unambiguous experimental signatures of altermagnetic superconductivity accessible via local measurements such as scanning tunneling microscopy and open unexplored pathways for designing tunable quantum devices.