The characteristic two-component blazar spectral energy distribution (SED) can be of either leptonic and/or hadronic origins. The potential association of the high-energy neutrino event IceCube-170922A with the flaring blazar TXS~0506+056 indicates that hadronic processes may operate in a blazar jet. Despite multi-wavelength follow-ups of the event and extensive theoretical modelings, the radiation mechanisms and the underlying magnetic field strength and configuration remain poorly understood. In this paper, we consider generic leptonic and hadronic blazar spectral models with distinct magnetic field strengths and radiation mechanisms. We analytically reproduce the SEDs and the neutrino flux of hadronic models, and predict their X-ray to $gamma$-ray polarization degrees. Furthermore, by performing relativistic magnetohydrodynamic (RMHD) simulations taking into account the polarization-dependent radiation transfer, we study the time-dependent multi-wavelength polarization variability of the proton synchrotron model under a shock scenario. Our results suggest that the high-energy polarization degree and the neutrino flux can be jointly used to pinpoint the leptonic and/or hadronic blazar radiation mechanisms in the X-ray and $gamma$-ray bands, and to infer the magnetic field strength in the emission region. Additionally, the temporal multi-wavelength polarization signatures in the proton synchrotron model shed light on the jet energy composition and the dynamical importance of magnetic fields in the blazar emission region. Future multi-wavelength polarimetry facilities such as {it IXPE} and {it AMEGO} together with neutrino telescopes such as {it IceCube} can provide unprecedented observational constraints to probe the blazar radiation mechanisms and jet dynamics.
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