  Entanglement witnesses based on first and second moments exist in the form of
spin-squeezing criteria for the detection of particle entanglement from
collective measurements, and in form of modified uncertainty relations for the
detection of mode entanglement or steering. By revealing a correspondence
between them, we show that metrologically useful spin squeezing reveals
multimode entanglement in symmetric spin states that are distributed into
addressable modes. We further derive tight state-independent multipartite
entanglement bounds on the spin-squeezing coefficient and point out their
connection to widely-used entanglement criteria that depend on the state's
polarization. Our results are relevant for state-of-the-art experiments where
symmetric entangled states are distributed into a number of addressable modes,
such as split spin-squeezed Bose-Einstein condensates.
