The goal of this article is to determine which closure model should be used in simulating complex flows of liquid-crystalline polymers (LCPs). We examine the performance of six closure models : the quadratic closure, a quadratic closure with finite molecular aspect ratio, the two Hinch-Leal closures, a hybrid between the quadratic and the first Hinch-Leal closures and a recently proposed Bingham closure. The first part of the article studies the predictions of the models; in homogeneous flows. We generate their bifurcation diagrams in the (U,Pe) plane, where U is the nematic strength and Pe is the Peclet number, and place special emphasis on the effects of the flow type. These solutions are then compared with the "exact solutions" of the unapproximated Doi theory. Results show the Bingham closure to give the best approximation to the Doi theory in terms of reproducing transitions between the director aligning, wagging and tumbling regimes at the correct values of Li and Re and predicting the arrest of periodic solutions by a mildly extensional flow. In the second part of the article, we employ the closure models to compute a complex flow in Bn eccentric cylinder geometry. All the models tested predict the same qualitative features of the LCP dynamics. Upon closer inspection of the quantitative differences among the solutions, the Bingham closure appears to be the most accurate. Based on these results, we recommend using the Bingham closure in simulating complex flows of LCPs.