We develop polynomial-time heuristic methods to solve unimodular quadratic programming (UQP) approximately, which is known to be NP-hard. In the UQP framework, we maximize a quadratic function of a vector of complex variables with unit modulus. Several problems in active sensing and wireless communication applications boil down to UQP. With this motivation, we present two new heuristic methods with polynomial complexity to solve the UQP approximately. The first method is called dominant-eigenvector-matching; here the solution is picked that matches the complex arguments of the dominant eigenvector of the Hermitian matrix in the UQP formulation. We also provide a performance guarantee for this method. The second heuristic method, a greedy strategy, is shown to provide a performance guarantee of (1 - 1/e) with respect to the optimal objective value given that the objective function possesses a property called string submodularity. We also present results from simulations to demonstrate the performance of these heuristic methods.