  How the objective everyday world emerges from the underlying quantum
behaviour of its microscopic constituents is an open question at the heart of
the foundations of quantum mechanics. Quantum Darwinism and spectrum broadcast
structure are two different frameworks providing key insight into this
question. Recent works, however, indicate these two frameworks can lead to
conflicting predictions on the objectivity of the state of a system interacting
with an environment. Here we provide a resolution to this issue by defining
strong quantum Darwinism and proving that it is equivalent to spectrum
broadcast structure when combined with strong independence of the
subenvironments. We further show that strong quantum Darwinism is sufficient
and necessary to signal state objectivity without the requirement of strong
independence. Our work unveils the deep connection between strong quantum
Darwinism and spectrum broadcast structure, thereby making fundamental progress
towards understanding and solving the emergence of classicality from the
quantum world. Together they provide us a sharper understanding of the
transition in terms of state structure, geometry, and quantum and classical
information.
