To date, all quantum mechanical experiments that have been performed agree with the assumption that quantum mechanical laws can be applied to any conceivable physical quantity. The working hypothesis of this talk will be that these laws can be extended to the notion of time, and, consequently, to the to the way in which two or more events can causally influence one another. This would have far-reaching consequences as our entire understanding of the world is based on the study of the causal relations between physical events. To begin, I will review some of the experimental demonstrations of indefinite causality in flat-spacetime carried out thus far. Firstly, I will present the first direct experimental demonstration of indefinite causality ever made, carried out by exploiting so-called "causal witnesses". Then I will extend the demonstration beyond quantum mechanics, presenting experimental demonstrations of indefinite causality with wider validity (i.e., valid for a larger class of so-called "generalised probabilistic theories"). In the second part of my talk, I will extend my study to the case of quantum superpositions of thermodynamic time’s arrows. To do this, I will leverage on the second law of thermodynamics, which allows one to associate a positive (negative) entropy variation in a thermodynamic process with the temporal "forward" ("time-reversal") direction. This will allow me to introduce the idea that quantum mechanics may permit quantum superpositions between thermodynamic processes yielding two opposite entropy variations. This would enable the existence of processes with a genuinely indefinite thermodynamic time's arrow. I will finally conclude by presenting some open questions and ideas for future studies.