It is demonstrated that the TaN molecule is the best candidate to search for a T,P-violating nuclear magnetic quadrupole moment (MQM), and it looks promising for the search for other T,P-odd effects.We report results of coupled-cluster calculations of T,P-odd effects in TaN produced by the Ta nucleus MQM, electron electric dipole moment (EDM), and scalar-pseudoscalar nucleus-electron interactions, as well as of the molecule-axis hyperfine structure constant and dipole moment. Nuclear calculations of 181Ta MQM are performed to express the T,P-odd effect in terms of the strength constants of T,P-odd nuclear forces, proton and neutron EDM, QCD parameter θ, and quark chromo-EDM.