  Chains of magnetic adatoms on superconductors have been discussed as
promising systems for realizing Majorana end states. Here, we show that dilute
Yu-Shiba-Rusinov (YSR) chains are also a versatile platform for quantum
magnetism and correlated electron dynamics, with widely adjustable spin values
and couplings. Focusing on subgap excitations, we derive an extended $t-J$
model for dilute quantum YSR chains and use it to study the phase diagram as
well as tunneling spectra. We explore the implications of quantum magnetism for
the formation of a topological superconducting phase, contrasting it to
existing models assuming classical spin textures.
