Lack of measurement independence can simulate quantum correlation even when signaling cannot

In 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, Bell scenario has been generalized into `semi-quantum' scenario where external quantum inputs are provided to the parties. We show that in `semi-quantum' scenario, entangled states produce correlations whose deterministic explanation is possible only if measurement independence is reduced. Thus in simulating quantum correlation `semi-quantum' scenario reveals a qualitative distinction between signaling and measurement dependence which is absent in Bell scenario. We further show that such distinction is not observed in `steering game' scenario, a special case of `semi-quantum' scenario.