Quantum complementarity <i>ad infinitum</i> : Switching higher-order coherence from infinity to zero

We reveal a sharp form of complementarity in the Janus state: a coherent superposition of two squeezed vacua. Using a compact generating-function formalism, we derive closed-form expressions for ${g}^{(k)}$ at all orders and introduce squeezing polynomials ${P}_{k}(z)$ that factorize the moments and expose phase-dependent scaling. The Janus state operates as a phase-tunable quantum switch for multiphoton correlations: when which-path or which-state information is available (trivial interference), the higher-order coherences ${g}^{(k)}$ reproduce the single-squeezed limit and diverge; when path information is erased and genuine interference from coherent superposition is enabled, multiphoton events are extinguished, driving ${g}^{(k)}\ensuremath{\rightarrow}0$ for all $k\ensuremath{\ge}3$. We obtain a hierarchy of suppression with exponents set by the interference phase and the parity of $k$, and we visualize the mechanism via pronounced Wigner-function negativity. The framework is general and experimentally accessible, enabling engineered nonclassical light with tailored higher-order statistics.