  We define bounded generation for $E_n$-algebras in chain complexes and prove
that for $n \geq 2$ this property is equivalent to homological stability. Using
this we prove a local-to-global principle for homological stability, which says
that if an $E_n$-algebra $A$ has homological stability (or equivalently the
topological chiral homology $\int_{R^n} A$ has homology stability), then so has
the topological chiral homology $\int_M A$ of any connected non-compact
manifold $M$. Using scanning, we reformulate the local-to-global homological
stability principle in a way that also applies to compact manifolds. We also
give several applications of our results
