Lack of measurement independence can simulate quantum correlations even when signaling can not

In the Bell scenario, any nonlocal correlation shared between two spatially separated parties can be modeled deterministically either by allowing communications between the two parties or by restricting their free will in choosing the measurement settings. Recently, the Bell scenario has been generalized into a semiquantum scenario where external quantum inputs are provided to the parties. We show that in the semiquantum scenario, entangled states produce correlations whose deterministic explanation is possible only if measurement independence is reduced. Thus in simulating quantum correlation the semiquantum scenario reveals a qualitative distinction between signaling and measurement dependence which is absent in the Bell scenario. We further show that such distinction is not observed in the steering-game scenario, a special case of the semiquantum scenario.