Hardy and Cabello Arguments in Spatial and Temporal Frauchiger-Renner Scenarios

We investigate Hardy- and Cabello-type logical structures within spatial and temporal extensions of the Frauchiger–Renner (FR) framework, embedding these constructions directly into the FR multi-observer architecture. In the spatial multi-observer scenario, both Hardy and Cabello contradictions arise, with the Cabello construction yielding the stronger violation,\(Δ_{\rm Cabello}^{\max}=0.1078\), which exceeds the maximal Hardy probability \(P_{H}^{\max}=\frac{5\sqrt{5}-11}{2}\approx 0.09017\). We then develop a sequential temporal FR protocol based on coherent multi-observer measurements performed on a single spin-$\frac{1}{2}$ system. In this temporal setting, the Hardy contradiction disappears identically due to dynamical constraints imposed by sequential state updates, whereas a finite Cabello-type violation survives, ${\mathrm{\Delta }}_{\mathrm{C}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{o}}^{\max }\approx 0.0674$. Our results establish a fundamental structural distinction between spatial entanglement and temporal multi-observer correlations in FR-type logical scenarios, and demonstrate that certain observer-independent description failures persist even without spacelike separation.