John,
You wrote, "Causality is energy transfer, not sequence." Your statement would only make sense if we didn't know that the transfer of heat energy from hot to cold causes water to boil. That's a definite ordered sequence of events.
In any case, you're missing the authors' point. They're talking about the transfer -- not of energy -- rather, of information contained in a disordered state, to a measurement of ordered, i.e., computable, results. A pot of water with a fire under it can be measured (with a thermometer) for energy content, and we have perfect knowledge of the ordered, covariant and continuous relation between the energy input and measurement output. What we don't know, is how the discrete states of observers inside the pot relate to the computable function we observe outside the pot. If Alice, in one region of the pot, sends Bob, in another region of the pot, a message about how fast she's moving, by the time Bob gets the message, any correlation between that information and Bob's state of motion will have been destroyed. As made clear in the article, even if Alice and Bob conspire in advance to send each other a message only when a set speed is achieved, their results will correlate -- at most -- only 75% of the time. Random correlations are limited to 50%.
What the authors propose is that the indeterminate information in a superposition of states can somehow be exploited such that a quantum "thermometer" -- i.e., the measurement output as a computable function -- can reach higher levels of information reliability, as the authors say, " ... the idea of indefinite causal structure brings the phenomenon of superposition into a new realm, the realm of the ordering of computational operations."
I approached the problem from a different perspective in my ICCS 2007 paper . Instead of superposition, I proposed applying the topological property of simple connectedness, and the complex systems property of self similarity: "3.7 The independence of time metrics in an n-dimensional system where time flows on a self avoiding random walk satisfies the multi-scale variety requirement. What we mean, is that the connectedness of the network is preserved in self-similar components that perform cooperative functions independent of the observed state of the system. Subsystems are self delimiting. Thereby, an analytically continuous complex system is tractable to analysis using the tools of discrete functions. This is an obvious crucial requirement for computability."
Tom