  The nonlocal set has received wide attention over recent years. Shortly
before, Li and Wang arXiv:2202.09034 proposed the concept of a locally stable
set: the only possible orthogonality preserving measurement on each subsystem
is trivial. Locally stable sets present stronger nonlocality than those sets
that are just locally indistinguishable. In this work, we focus on the
constructions of locally stable sets in multipartite quantum systems. First,
two lemmas are put forward to prove that an orthogonality-preserving local
measurement must be trivial. Then we present the constructions of locally
stable sets with minimum cardinality in bipartite quantum systems
$\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ $(d\geq 3)$ and
$\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}$ $(3\leq d_{1}\leq d_{2})$.
Moreover, for the multipartite quantum systems $(\mathbb{C}^{d})^{\otimes n}$
$(d\geq 2)$ and $\otimes^{n}_{i=1}\mathbb{C}^{d_{i}}$ $(3\leq d_{1}\leq
d_{2}\leq\cdots\leq d_{n})$, we also obtain $d+1$ and $d_{n}+1$ locally stable
orthogonal states respectively. Fortunately, our constructions reach the lower
bound of the cardinality on the locally stable sets, which provides a positive
and complete answer to an open problem raised in arXiv:2202.09034 .
