Quantum Complementarity ad Infinitum: Switching Higher-Order Coherence from Infinity to Zero

We report a profound manifestation of quantum complementarity in the higher-order photon statistics of the ``Janus state,'' a coherent superposition of two squeezed vacua. We find that the state acts as a perfect quantum switch for multi-photon correlations, toggled by the availability of which-path information. Erasing this information activates quantum interference that can be tuned to be maximally destructive. This reveals a remarkable hierarchy of suppression: while two-photon correlations remain finite, we prove analytically and demonstrate numerically that it is possible to drive all higher-order correlations ($g^{(k)}$ for $k \ge 3$) to zero. This transition from the extreme bunching of the constituent states ($g^{(k)} \to \infty$) to a state of profound quantum order is visualized by the emergence of negativity in the state's Wigner function, an unambiguous signature of non-classicality. This work provides a foundational demonstration of quantum complementarity in multi-photon statistics and introduces a new paradigm for engineering highly ordered, non-classical light from Gaussian resources.