  We consider the characterization of entanglement depth in a quantum many-body
system from the device-independent perspective; i.e., certifying how many
particles are genuinely entangled without relying on assumptions on the system
itself nor on the measurements performed. We obtain device-independent
witnesses of entanglement depth (DIWEDs) using the Bell inequalities introduced
in [J. Tura et al., Science 344, 1256 (2014)] and compute their
$k$-producibility bounds. To this end, we present two complementary methods:
First, a variational one, yielding a possibly optimal $k$-producible state.
Second, a certificate of optimality via a semi-definite program, based on a
relaxation of the quantum marginal problem. Numerical results suggest a clear
pattern on $k$-producible bounds for large system sizes, which we then tackle
analytically in the thermodynamic limit. Contrary to existing DIWEDs, the ones
we present here can be effectively measured by accessing only collective
measurements and second moments thereof. These technical requirements are met
in current experiments, which have already been performed in the context of
detecting Bell correlations in quantum many-body systems of $5\cdot 10^2 \sim 5
\cdot 10^5$ atoms.
