  How to detect quantum correlations in bi-partite scenarios using a split
many-body system and collective measurements on each party? We address this
question by deriving entanglement witnesses using either only first or first
and second order moments of local collective spin components. In both cases, we
derive optimal witnesses for spatially split spin squeezed states in the
presence of local white noise. We then compare the two optimal witnesses with
respect to their resistance to various noise sources operating either at the
preparation or at the detection level. We finally evaluate the statistics
required to estimate the value of these witnesses when measuring a split
spin-squeezed Bose-Einstein condensate. Our results can be seen as a step
towards Bell tests with many-body systems.
