Causal modelling is a tool for generating causal explanations of observed correlations and has led to a deeper understanding of correlations in quantum networks. Existing frameworks for quantum causality tend to focus on acyclic causal structures that are not fine-tuned i.e., where causal connections between variables necessarily create correlations between them. However, fine-tuned causal models which permit causation without correlation, play a crucial role in cryptography, and cyclic causal models can be used to model physical processes involving feedback and may also be relevant in exotic solutions of general relativity. Here we develop a causal modelling framework capable of modelling causation in these general scenarios. The key feature of our framework is that it allows operational and relativistic notions of causality to be independently defined and for connections between them to be established. The framework first gives an operational way to study causation that allows for cyclic, fine-tuned and non-classical causal influences. We then consider how a causal model can be embedded in a space-time structure (modelled as a partial order) and propose a compatibility condition for ensuring that the embedded causal model does not allow signalling outside the space-time future. We identify several distinct classes of causal loops that can arise in our framework, showing that compatibility with a space-time can rule out only some of them. We discuss conditions for preventing superluminal signalling within arbitrary (and possibly cyclic) causal structures and consider models of causation in post-quantum theories admitting so-called jamming correlations. Finally, this work introduces the concept of a "higher-order affects relation", which is useful for causal discovery in fined-tuned causal models. ContentsI. Introduction II. Preliminaries: Acyclic and faithful causal models III. Motivation for analysing fine-tuned and cyclic causal models A. Friedman's thermostat and the one-time pad B. Jamming non-local correlations IV. The framework, Part 1: Causality A. Cyclic and fine-tuned causal models B. Interventions and affects relations C. Conditional and higher-order affects relations D. Relationships between concepts V. The framework, Part 2: Space-time A. Space-time structure B. Embedding of a causal model in a space-time structure C. Compatibility of a causal model with an embedding in space-time D. Necessary and sufficient conditions for compatibility VI. Causal loops and their space-time embeddings A. Different classes of causal loops B. Possibility of compatibly embedding causal loops in space-time

Causality is fundamental to science, but it appears in several different forms. One is relativistic causality, which is tied to a space-time structure and forbids signalling outside the future. A second is an operational notion of causation that considers the flow of information between physical systems and interventions on them. In In [Vilasini and Colbeck, Phys. Rev. A. xx (2022)], we propose a framework for characterising when a causal model can coexist with relativistic principles such as no superluminal signalling, while allowing for cyclic and non-classical causal influences and the possibility of causation without signalling. In a theory without superluminal causation, both superluminal signalling and causal loops are not possible in Minkowski space-time. Here we demonstrate that if we only forbid superluminal signalling, superluminal causation remains possible and show the mathematical possibility of causal loops that can be embedded in a Minkowski space-time without leading to superluminal signalling. The existence of such loops in the given space-time could in principle be operationally verified using interventions. This establishes that the physical principle of no superluminal signalling is not by itself sufficient to rule out causal loops between Minkowski space-time events. Interestingly, the conditions required to rule out causal loops in a space-time depend on the dimension. Whether such loops are possible in three spatial dimensions remains an important open question.Introduction.-Understanding cause-effect relations is central to the scientific method, yet there are several inequivalent notions of causality. Often, it is defined with respect to a background space-time structure, after which causal structure and space-time structure are treated synonymously. An alternative is to define causality operationally and independently of space-time. One way to do this is through causal models, which are based on intervening on physical systems and analysing the resulting correlations [1,2]. This is the approach we take here. Causal models have been extensively applied to situations involving classical variables, being used for instance for medical testing [3,4], economic predictions [1, 5], and machine learning [6][7][8].

Relativistic protocols have been proposed to overcome certain impossibility results in classical and quantum cryptography. In such a setting, one takes the location of honest players into account, and uses the signalling limit given by the speed of light to constraint the abilities of dishonest agents. However, composing such protocols with each other to construct new cryptographic resources is known to be insecure in some cases. To make general statements about such constructions, a composable framework for modelling cryptographic security in Minkowski space is required. Here, we introduce a framework for performing such a modular security analysis of classical and quantum cryptographic schemes in Minkowski space. As an application, we show that (1) fair and unbiased coin flipping can be constructed from a simple resource called channel with delay; (2) biased coin flipping, bit commitment and channel with delay through any classical, quantum or post-quantum relativistic protocols are all impossible without further setup assumptions; (3) it is impossible to securely increase the delay of a channel, given several short-delay channels as ingredients. Results (1) and (3) imply in particular the non-composability of existing relativistic bit commitment and coin flipping protocols.