How Difficult Is It to Prove the Quantumness of Macroscropic States?

General wisdom tells us that if two quantum states are macroscopically distinguishable then their superposition should be hard to observe. We make this intuition precise and general by quantifying the difficulty to observe the quantum nature of a superposition of two states that can be distinguished without microscopic accuracy. First, we quantify the distinguishability of any given pair of quantum states with measurement devices lacking microscopic accuracy, i.e., measurements suffering from limited resolution or limited sensitivity. Next, we quantify the required stability that has to be fulfilled by any measurement setup able to distinguish their superposition from a mere mixture. Finally, by establishing a relationship between the stability requirement and the distinguishability with inaccurate measurements of the two superposed states, we demonstrate that, indeed, the more distinguishable the states are, the more demanding the stability requirements.