  In this paper, we investigate the hierarchical structure of the $n$-partite
quantum states. We present a whole set of hierarchical quantifications as a
method of characterizing quantum states, which go beyond genuine multipartite
entanglement measures and allow for fine identification among distinct
entanglement contributions. This kind of quantifications, termed $k$-GM
concurrence, can unambiguously classify entangled states into $(n-1)$ distinct
classes from the perspective of $k$-nonseparability with $k$ running from $n$
down to 2, and comply with the axiomatic conditions of an entanglement measure.
Compared to $k$-ME concurrence
[\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.062323} {Phys.
Rev. A \textbf{86}, 062323 (2012)}], the hierarchical measures proposed by us
embody advantages in distinguishing same class entangled state and measuring
continuity. In addition, we establish the relation between $k$-ME concurrence
and $k$-GM concurrence, and further derive a strong lower bound on the $k$-GM
concurrence by exploiting the permutationally invariant part of a quantum
state. Furthermore, we parametrize $k$-GM concurrence to obtain two more
general and complete categories of quantifications, $q$-$k$-GM concurrence
$(q>1)$ and $\alpha$-$k$-GM concurrence $(0\leq\alpha<1)$, which obey the
properties enjoyed by $k$-GM concurrence as well. In particular,
$\alpha$-$2$-GM concurrence $(0<\alpha<1)$ determines that the GHZ state and
the $W$ state belong to the same hierarchy, and it is proven in detail
satisfying the requirement that the GHZ state is more entangled than the $W$
state in multiqubit systems.
