Quantum network nonlocality plays a crucial role in long-distance quantum communication, and nonlocality sharing via sequential measurements provides a novel perspective for large-scale quantum network applications. In this paper, we investigate network nonlocality sharing in the two-forked tree-shaped network. For the two-input $n$-layer $(n\ge 2)$ scenario, the number of sequential observers in existing schemes is constrained by the layer $n$ and the number of sequential sides. Here we achieve unbounded sharing of network nonlocality for arbitrary $n$ by introducing sharpness parameters into Bell state measurements and allowing sequential observers to perform measurements with different sharpness parameters. An unbounded number of independent observers on each of the $m(m=1,\cdots ,2^{n-2})$ sequential sides can simultaneously share non-$(2^n-2)$-locality by violating the two-input $(2^n-2)$-local inequality with all other nonsequential observers. Furthermore, we extend the two-input scenario to a generalized $d$-input $(d\ge 2)$ case, investigating how the input number $d$ affects network nonlocality sharing. A general model for sequential measurement scenarios in $d$-input two-forked $n$-layer tree-shaped networks is established. Using the simplest case $(n=3)$ as an example, we investigate non-six-locality sharing based on the $d$-input six-local inequality and establish the relationship between the number of sequential observers, the number of sequential sides, and the input $d$. Based on the results, we conjecture that the two-input configuration is optimal for nonlocality sharing in two-forked tree-shaped networks.

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• Nonlocality
• Quantum correlations in quantum information
• Quantum foundations
• Quantum measurements
• Quantum networks