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This study focuses on tensor <i>Z</i>-eigenvalue localization and its application in the geometric measure of entanglement for multipartite quantum states. A new <i>Z</i>-eigenvalue localization theorem and the bounds for the Z-spectral radius are derived, which are more precise than some of the existing results. On the other hand, we present theoretical bounds of the geometric measure of entanglement for a weakly symmetric multipartite quantum state with non-negative amplitudes by virtue of different distance measures. Numerical examples show that these conclusions are superior to the existing results in quantum physics in some cases.