Blazar Variability as a Probe of Ultra-High-Energy Cosmic Rays and Very-High-Energy Gamma Rays: A Multi-Messenger Review Across All Timescales

Blazars are variable at every timescale that has been observed, from minute-scale flares to secular trends spanning decades, and across the whole electromagnetic spectrum from radio to very-high-energy (VHE; Eγ≳100 GeV) γ-rays. This variability is not merely a nuisance to be averaged over: it is a time-resolved record of the physical conditions in the jet, and therefore a diagnostic of whether, and how, these sources accelerate particles to the highest energies. This review takes variability itself as its organizing principle. Rather than asking whether a particular blazar subclass accelerates ultra-high-energy cosmic rays (UHECRs), we ask what the amplitude, timescale, spectral evolution, inter-band correlation, and polarization behavior of blazar variability reveal about the underlying particle acceleration, and how that information bears on the production of VHE γ-rays and the acceleration of UHECRs. We organize the discussion by timescale, from microvariability through secular evolution, and by messenger, treating the electromagnetic light curves, the high-energy neutrino signal, and the (necessarily time-averaged) UHECR flux as complementary probes of the same accelerator. We review how causality arguments applied to rapid flares constrain the size, Doppler factor, and magnetic field of the emission region; how the presence or absence of inter-band correlation discriminates between leptonic and lepto-hadronic scenarios; and why the enormous magnetic time delays of charged cosmic rays sever the temporal link that neutrinos preserve. We close with the prospects that the Cherenkov Telescope Array Observatory (CTAO), the Southern Wide-field Gamma-ray Observatory (SWGO), IceCube-Gen2, and the AugerPrime upgrade open for turning time-domain information into the identification of the astrophysical accelerators of the most energetic particles in the Universe.