Abstract: The stress-strain relations of saturated sand are anisotropic and state-dependent, thus the initiation of its static liquefaction relies on stress paths. While various criteria have been proposed to predict the triggering of static liquefaction, their accuracy is normally examined under triaxial stress paths. When subjected to complex stress paths that involve the rotation of the principal stress directions and different relative magnitudes of the intermediate principal stress, the accuracy of these criteria remains to be an open question. Here the capacity of three criteria is evaluated, including the second-order work, symmetric part of the elastoplastic stiffness matrix and instability modulus, for predicting the static liquefaction under complex stress paths, by employing a state-dependent, anisotropic constitutive model and comparing against the hollow cylindrical torsional shear tests on Toyoura sand. It is found that: (1) The instability condition derived from the elastoplastic stiffness matrix does not depend on the stress paths, thus being more general, and the predicted liquefaction initiation is earlier than or coincident with the actual instability point. (2) The instability modulus approach can predict the initiation of static liquefaction, however, the instability conditions vary across different loading paths. The instability line is obtained, through which the influences of factors like the intermediate principal stress and the principal stress directions on the peak mobilized friction angle before liquefaction are studied.