  Experimental detection of entanglement certainly disturbs the underlying
shared state. It is possible that the entanglement content of the system is
lost in the process of its detection. This observation has led to the study of
sequential detection properties of various quantum correlations. Here, we
investigate the sequential detection of genuinely multipartite entanglement of
quantum systems composed of an arbitrary number of qubits. In order to detect
genuine multipartite entanglement sequentially, observers can recycle any fixed
subset of all the qubits, thus leading to a hierarchy of scenarios, categorized
according to the number of qubits which are recycled by the observers. We show
that the sequential detection of genuine multipartite entanglement, for every
scenario in the hierarchy, leads to an unboundedly long sequence. This is shown
to be possible if the initial state shared among the observers is the
multipartite generalized Greenberger-Horne-Zeilinger state and is a class of
mixed states. A comparison among different hierarchical scenarios is drawn
based on the number of sequential detections of genuine multipartite
entanglement for a specific measurement strategy employed by observers.
