  We propose and analyze a novel mechanism for long-range spin-spin
interactions in diamond nanostructures. The interactions between electronic
spins, associated with nitrogen-vacancy centers in diamond, are mediated by
their coupling via strain to the vibrational mode of a diamond mechanical
nanoresonator. This coupling results in phonon-mediated effective spin-spin
interactions that can be used to generate squeezed states of a spin ensemble.
We show that spin dephasing and relaxation can be largely suppressed, allowing
for substantial spin squeezing under realistic experimental conditions. Our
approach has implications for spin-ensemble magnetometry, as well as
phonon-mediated quantum information processing with spin qubits.
