  Despite a surge of interest in submodular maximization in the data stream
model, there remain significant gaps in our knowledge about what can be
achieved in this setting, especially when dealing with multiple constraints. In
this work, we nearly close several basic gaps in submodular maximization
subject to $k$ matroid constraints in the data stream model. We present a new
hardness result showing that super polynomial memory in $k$ is needed to obtain
an $o(k / \log k)$-approximation. This implies near optimality of prior
algorithms. For the same setting, we show that one can nevertheless obtain a
constant-factor approximation by maintaining a set of elements whose size is
independent of the stream size. Finally, for bipartite matching constraints, a
well-known special case of matroid intersection, we present a new technique to
obtain hardness bounds that are significantly stronger than those obtained with
prior approaches. Prior results left it open whether a $2$-approximation may
exist in this setting, and only a complexity-theoretic hardness of $1.91$ was
known. We prove an unconditional hardness of $2.69$.
