Matrix (or operator) recovery from linear measurements is a well-studied problem. However, there are situations where only bilinear or quadratic measurements are available. A bilinear or quadratic problem can easily be transformed into a linear one, but it raises questions when the linearized problem is solvable and what is the cost of linearization. In this work, we study a few specific cases of this general problem and show when the bilinear problem is solvable. Using this result and certain properties of polynomial rings, we present a scenario when the quadratic problem can be linearized at the cost of just a linear number of additional measurements. Finally, we link our results back to two applications that inspired it: Time Encoding Machines and Continuous Localisation.