  Identifying the $k$-partite entanglement and $k$-nonseparability of general
$N$-partite quantum states are fundamental issues in quantum information
theory. By use of computable inequalities of nonlinear operators, we present
some simple and powerful $k$-partite entanglement and $k$-nonseparability
criteria that works very well and allow for a simple and inexpensive test for
the whole hierarchy of $k$-partite entanglement and $k$-separability of
$N$-partite systems with $k$ running from $N$ down to 2. We illustrate their
strengths by considering several examples in which our criteria perform better
than other known detection criteria. We are able to detect $k$-partite
entanglement and $k$-nonseparabilty of multipartite systems which have
previously not been identified. In addition, our results can be implemented in
today's experiments.
