Equivariant quantum Boolean functions and quantum locality tests

We study quantum Boolean functions on $ \mathbb{C}[\mathbb{F}_2^{2n}] $ that are equivariant under the natural action of $ S_3\times S_n $, where $ S_3 $ permutes the three nonzero elements of $ \mathbb{F}_2^2 $, and $ S_n $ permutes the coordinate positions $ n $. Using character theory, we decompose the underlying representation into irreducible $ S_3\times S_n $ sectors and show that equivariant operators reduce to blocks acting on finite-dimensional multiplicity spaces. Based on this reduction, we construct a promise based quantum verification protocol combining a group symmetry test, a local purity test, and an energy expectation test. The protocol has completeness and promise gap soundness at least $ 1-\delta $, where $ \delta\in(0, 1) $. If the three bounded statistics are estimated to additive accuracies proportional to the corresponding promise gaps, its total query complexity is $ O\ ((\alpha_{\mathrm{sym}}^{-2} + \alpha_{\mathrm{loc}}^{-2} + \alpha_{\mathrm{eng}}^{-2}) \log{\delta^{-1}}), $ up to explicit group sampling, locality, subsystem operation, and copy counting factors, where $ \alpha_{\mathrm{sym}} $, $ \alpha_{\mathrm{loc}} $, $ \alpha_{\mathrm{eng}}\in (0, 1) $. Conditional distance soundness follows when the stated error bound property relating the trace distance to the three promise gaps holds. The resulting framework provides a symmetry reduced approach to quantum locality verification without requiring informationally complete state tomography.