Common filters are usually based on the linear approximation of the optimal minimum mean square error estimator. The Extended and Unscented Kalman Filters handle nonlinearity through linearization and unscented transformation, respectively, but remain linear estimators, meaning that the state estimate is a linear function of the measurement. This paper proposes a quadratic approximation of the optimal estimator, creating the Quadratic Extended and Quadratic Unscented Kalman Filter. These retain the structure of their linear counterpart, but include information from the measurement square to obtain a more accurate estimate. Numerical results show the benefits in accuracy of the new technique, which can be generalized to upgrade other linear estimators to their quadratic versions.