Superpositions of Gaussian states, including circular states and generalized circular states, exhibit a rich variety of nonclassical features such as Wigner negativity and sub-Planck phase-space structures. The Tsirelson probability, central to the Tsirelson precession protocol, is defined as the average probability that a precessing quadrature yields a positive outcome when measured at equally spaced times, with the classical bound given by $1/2 \pm 1/(2d).$ In this work, we compute this probability for superpositions of Gaussian states. For circular states (superpositions of $d$ coherent states), we analytically calculate their Tsirelson probability and find violations of the classical bounds for $d=3,5,7$. For generalized circular states, which incorporate squeezing, we derive a general expression and identify parameter regimes that enhance the violation. Our results provide a comprehensive map of the dynamical nonclassicality of superpositions of Gaussian states and establish them as versatile platforms for testing nonclassicality witnesses.
Gaussian states are fundamental in continuous-variable quantum information, yet characterizing non-Gaussianity remains challenging due to the non-convexity of the Gaussian set. Existing witnesses typically rely on Wigner negativity or other information-theoretic quantities. In this work, we develop a group-theoretic, multi-copy approach to detect non-Gaussianity in bosonic systems. We study passive linear optical transformations that mix copies of a quantum state and analyze their commutation with identical Gaussian unitaries applied to each copy. Orthogonal copy-mixing transformations commute with the symplectic part of the Gaussian action, while the displacement part restricts the symmetry to the stabilizer of the collective mode. This structure yields a family of witnesses satisfied by all single-mode Gaussian states. Fixing the thermal reference parameter via the purity, violation of these identities certifies non-Gaussianity. We illustrate the method with several single-mode examples and present an experimental protocol based on passive interferometry and photon-number-resolved detection, showing that the relevant multi-copy expectation values can be estimated from bounded phase observables. Finally, we extend the construction to multi-mode systems and discuss how the same symmetry framework may lead to quantitative measures of non-Gaussianity.