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The detection of the high-energy ($\sim290$ TeV) neutrino coincident with theflaring blazar TXS 0506+056, the first and only $3\sigma$ neutrino-sourceassociation to date, provides new, multimessenger tests of the weak equivalenceprinciple (WEP) and Lorentz invariance. Assuming that the flight timedifference between the TeV neutrino and gamma-ray photons from the blazar flareis mainly caused by the gravitational potential of the Laniakea supercluster ofgalaxies, we show that the deviation from the WEP for neutrinos and photons isconservatively constrained to have an accuracy of $10^{-6}-10^{-7}$, which is3--4 orders of magnitude better than previous results placed by MeV neutrinosfrom supernova 1987A. In addition, we demonstrate that the association of theTeV neutrino with the blazar flare sets limits on the energy scales of quantumgravity for both linear and quadratic violations of Lorentz invariance (LIV) to$E_{\rm QG, 1}>3.2\times10^{15}-3.7\times10^{16}$ GeV and $E_{\rm QG,2}>4.0\times10^{10}-1.4\times10^{11}$ GeV. These improve previous limits onboth linear and quadratic LIV energy scales in neutrino propagation by 5--7orders of magnitude.

Multimessenger Tests of Einstein's Weak Equivalence Principle and Lorentz Invariance with a High-energy Neutrino from a Flaring Blazar