GHZ State

The GHZ state is an entangled state. Most often, the GHZ state is defined for three qubits and is given by

$$|\mathrm{GHZ}⟩=\frac{|000⟩+|111⟩}{\sqrt{2}}.$$

The $N$-qubit generalization of the GHZ state is

$$|\mathrm{GHZ}⟩=\frac{|0{⟩}^{\otimes N}+|1{⟩}^{\otimes N}}{\sqrt{2}}.$$

Relation to Cluster states

The GHZ state is not a cluster state, but it can be made into a cluster state by local Clifford operations. To see this, consider applying a Hadamard gate to all qubits except the first:

$$\prod _{i=1}^N{\hat{H}}_i|\mathrm{GHZ}⟩=\frac{|0+\dots +⟩+|1-\dots -⟩}{\sqrt{2}}$$

The generators become:

$$\begin{array}{r}{\hat{X}}_1\dots \widehat{Z_N}\\ {\hat{Z}}_1\widehat{X_2}\\ {\hat{Z}}_1\widehat{X_3}\\ ⋮\\ {\hat{Z}}_1\widehat{X_N}\end{array}$$

which are the stabilizers corresponding to a star shaped graph state.

References

https://quantumcomputing.stackexchange.com/questions/30050/difference-between-a-star-graph-state-and-ghz-graph-state