  Condensed matter is composed of a small set of identical units, yet it shows
an immense range of behaviour. Recently, an array of cold atoms was used to
generate long-range quantum entanglement, a property of topological matter.
Another approach to strong non-local correlations employs the macroscopic
coherence of superconductors. Impurity spins in superconductors are thought to
be unamenable to the formation of long-range spin entanglement because each
spin tends to be screened by binding to a quasiparticle from the superconductor
to form a local singlet. Here we demonstrate that it is possible to attach a
second quasiparticle to the spin, overscreening it into a doublet state
carrying ferromagnetic correlations between two quasiparticles over a
micrometer distance. To demonstrate this effect, which is strongest for equal
binding, we symmetrically couple the spin of a quantum dot to two ultrasmall
superconducting islands. The overscreened state requires sufficiently large
Coulomb repulsion and exchange binding to become well defined. We predict that
this state will carry long-range correlations for an alternating chain of
quantum dots and superconducting islands, opening a new route to controllable
large-scale entanglement in the solid state.
