2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/213850In this paper, we show that there are eight distinct forms of the Greenberger-Horne-Zeilinger (GHZ) argument for the four-qubit cluster state $|ϕ_4>$ and forty eight distinct forms for the five-qubit cluster state $|ϕ_5>$ in the case of the one-dimensional lattice. The proof is obtained by regarding the pair qubits as a single object and constructing the new Pauli-like operators. The method can be easily extended to the case of the N-qubit system and the associated Bell inequalities are also discussed. Consequently, we present a complete construction of the GHZ theorem for the cluster states of N-qubit in the case of the one-dimensional lattice.7 pages, 2 figures, We believe the paper may be of particular interest to the readers of your journal because the study it reports stated a complete construction of the GHZ theorem for the cluster states of N-qubit in the case of the one-dimensional latticeQuantum Physicstext