Temporal nonlocality of a qudit resides in the input state, not the channel, and certifies temporal teleportation up to a fundamental limit

Correlations between two moments in time can be too strong for any classical explanation – and, remarkably, this can happen for a single quantum system measured twice, with no second particle involved. We show that when one qudit is sent through a noisy channel, the strength of this “nonlocality in time” – the temporal nonlocality robustness $\mathrm{T}\mathrm{N}\mathrm{R}$ – is carried entirely by the starting state: it vanishes precisely when the input is maximally mixed (completely random), $\mathrm{T}\mathrm{N}\mathrm{R}({\rho }_A,\mathcal{E})=0\iff {\rho }_A=\mathbb{1}\mathrm{/}d$, for the standard noise families. The resource is not any coherence in the channel but the back-action of the input’s mixedness, and it survives even complete decoherence. This is at once a power and a trap. As a power, $\mathrm{T}\mathrm{N}\mathrm{R}$ device-independently lower-bounds the fidelity of temporal teleportation – sending an unknown state forward in time – reaching $7\mathrm{/}9$ at $d=3$, without trusting the measuring devices. As a trap, because the certified quantity is decoupled from the channel’s actual coherence transmission, it can certify more than the channel delivers: an injective (reversible) unitary attains the maximal temporal-Bell signal yet teleports below the classical baseline. We resolve this over-certification completely – a universal cap $\mathrm{T}\mathrm{N}\mathrm{R}\le (d-1)\mathrm{/}d$ with an exact channel-resolved value, honest certification for the depolarizing channel and for any sufficiently mixed probe, and a proof that no choice of probes makes it channel-universal. Underpinning the results is a unified semidefinite-programming hierarchy of the temporal entanglement, steering and nonlocality robustnesses ($\mathrm{T}\mathrm{E}\mathrm{R}$, $\mathrm{T}\mathrm{S}\mathrm{R}$, $\mathrm{T}\mathrm{N}\mathrm{R}$), with a strict lower hierarchy and an upper one conditional on no-signaling in time ($\mathrm{N}\mathrm{S}\mathrm{I}\mathrm{T}$). All structure is verified numerically for $d=2$ through $5$.