Quantum Complementarity ad Infinitum: 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 Pₖ(z) that factorize the moments and expose phase-dependent scaling. The Janus state operates as a phase-tunable quantum switch for multi-photon correlations: when which-path/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, multi-photon events are extinguished, driving g⁽ᵏ⁾\!→\!0 for all k\!≥\!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.