Fig. 1: Space-time diagrams of the causal influences needed, in every round of the Bell-like test, for each of the possible solutions to Shimony’s problem in light of our results.

In all diagrams, black arrows represent causal influences common to all possibilities. a Complete measurement dependence. It can occur in, essentially, two ways. The first is with complete superdeterminism without retrocausality. In this case, Alice and Bob do not have freedom of choice to choose the measurement settings, X and Y, respectively. Instead, the settings are determined by the distribution of the HVs in the past light cones of X and Y, represented by the lower node Λ. The causal influences between the lower Λ and X and Y are represented by violet continuous arrows. The second way complete MD can occur is with complete retrocausality with freedom of choice. In this case, Alice and Bob have freedom to choose the “nominal” measurement settings X and Y, but the actual measurements are determined by the distribution of the HVs in the future light cones of X and Y, represented by the upper node Λ. The causal influences between the upper Λ and X and Y are represented by violet dashed arrows. b Complete parameter dependence. X is decided by Alice and Y is decided by Bob. However, X does not only influence Alice’s measurement outcome, represented by A, but also Bob’s measurement outcome, represented by B, which is outside the light cone of X. This superluminal influence (or “action at a distance'') is represented by a red arrow. Similarly, Y does not only influence B, but also A. c Complete outcome dependence. X is decided by Alice and Y is decided by Bob. However, A and B are causally connected despite they are space-like separated. These superluminal influences are represented by blue arrows. d No hidden variables. X is decided freely by Alice and Y is decided freely by Bob. A is causally connected only to the quantum state, represented by Ψ, and X. Similarly, B is causally connected only to Ψ and Y.