  We derive a general framework to identify genuinely multipartite entangled
mixed quantum states in arbitrary-dimensional systems and show in exemplary
cases that the constructed criteria are stronger than those previously known.
Our criteria are simple functions of the given quantum state and detect genuine
multipartite entanglement that had not been identified so far. They are
experimentally accessible without quantum state tomography and are easily
computable as no optimization or eigenvalue evaluation is needed.
