Precision requirements for observing macroscopic quantum effects

It has recently been conjectured that detecting quantum effects such as superposition or entanglement for macroscopic systems always requires high measurement precision. Analyzing an apparent counterexample involving macroscopic coherent states and Kerr or higher-order nonlinearities, we find that while measurements with coarse outcomes can be sufficient, the phase control precision of the necessary nonlinear operations has to increase with the size of the system. This suggests a refined conjecture that either the outcome precision or the control precision of the measurements has to increase with system size.