We study the notion of causal orders for the cases of (classical and quantum) circuits and spacetime events. We show that every circuit can be immersed into a classical spacetime, preserving the compatibility between the two causal structures. Using the process matrix formalism, we analyse the realisations of the quantum switch using 4 and 3 spacetime events in classical spacetimes with fixed causal orders, and the realisation of a gravitational switch with only 2 spacetime events that features superpositions of different gravitational field configurations and their respective causal orders. We show that the current quantum switch experimental implementations do not feature superpositions of causal orders between spacetime events, and that these superpositions can only occur in the context of superposed gravitational fields. We also discuss a recently introduced operational notion of an event, which does allow for superpositions of respective causal orders in flat spacetime quantum switch implementations. We construct two observables that can distinguish between the quantum switch realisations in classical spacetimes, and gravitational switch implementations in superposed spacetimes. Finally, we discuss our results in the light of the modern relational approach to physics.
 J. S. Bell, Physics Physique Fizika 1, 195 (1964).
 L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. A. Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner and P. Walther, Nature Communications 6, 7913 (2015).
 L. Hardy, Journal of Physics A: Mathematical and Theoretical 40, 3081 (2007).
 M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi and Č. Brukner, New Journal of Physics 17, 102001 (2015).
 H. Lichtenegger and B. Mashhoon, in The Measurement of Gravitomagnetism: A Challenging Enterprise, edited by L. Iorio, 13, Nova Science Pub Inc, New York (2007).
 M. Blagojević, Gravitation and Gauge Symmetries, Institute of Physics Publishing, Bristol (2002).
 A. S. Eddington, Space time and gravitation, Cambridge University Press, Cambridge (1921).
 C. Misner, K. Thorne and J. Wheeler, Gravitation, W. H. Freeman, San Francisco (1973).
 C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge (2004).
 J. Janjić, N. Paunković and M. Vojinović, in preparation.
 C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory, Cambridge University Press, Cambridge (2014).
 R. A. Bertlmann and P. Krammer, Journal of Physics A: Mathematical and Theoretical 41, 235303 (2008).
 M. Trassinelli, "Conditional probabilities of measurements, quantum time, and the Wigner's-friend case", Physical Review A 105 3, 032213 (2022).
 Huan Cao, Jessica Bavaresco, Ning-Ning Wang, Lee A. Rozema, Chao Zhang, Yun-Feng Huang, Bi-Heng Liu, Chuan-Feng Li, Guang-Can Guo, and Philip Walther, "Semi-device-independent certification of indefinite causal order in a photonic quantum switch", Optica 10 5, 561 (2023).
 Marco Fellous-Asiani, Raphaël Mothe, Léa Bresque, Hippolyte Dourdent, Patrice A. Camati, Alastair A. Abbott, Alexia Auffèves, and Cyril Branciard, "Comparing the quantum switch and its simulations with energetically constrained operations", Physical Review Research 5 2, 023111 (2023).
 Ricardo Faleiro, Nikola Paunkovic, and Marko Vojinovic, "Operational interpretation of the vacuum and process matrices for identical particles", Quantum 7, 986 (2023).
 Aurélien Drezet, "Indefinite causal order with fixed temporal order for electrons and positrons", Quantum Studies: Mathematics and Foundations 10 1, 101 (2023).
 Emily Adlam, "Is there causation in fundamental physics? New insights from process matrices and quantum causal modelling", Synthese 201 5, 152 (2023).
 Kyrylo Simonov, Gianluca Francica, Giacomo Guarnieri, and Mauro Paternostro, "Work extraction from coherently activated maps via quantum switch", Physical Review A 105 3, 032217 (2022).
 Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard, "Quantum Circuits with Classical Versus Quantum Control of Causal Order", PRX Quantum 2 3, 030335 (2021).
 David Felce, Nicetu Tibau Vidal, Vlatko Vedral, and Eduardo O. Dias, "Indefinite causal orders from superpositions in time", Physical Review A 105 6, 062216 (2022).
 Natália S. Móller, Bruna Sahdo, and Nelson Yokomizo, "Quantum switch in the gravity of Earth", Physical Review A 104 4, 042414 (2021).
 Nikola Paunković and Marko Vojinović, "Equivalence Principle in Classical and Quantum Gravity", Universe 8 11, 598 (2022).
 Laurie Letertre, "Causal nonseparability and its implications for spatiotemporal relations", Studies in History and Philosophy of Science 95, 64 (2022).
 Nick Ormrod, Augustin Vanrietvelde, and Jonathan Barrett, "Causal structure in the presence of sectorial constraints, with application to the quantum switch", Quantum 7, 1028 (2023).
 Erickson Tjoa, "Quantum teleportation with relativistic communication from first principles", Physical Review A 106 3, 032432 (2022).
 Jian Wei Cheong, Andri Pradana, and Lock Yue Chew, "Communication advantage of quantum compositions of channels from non-Markovianity", Physical Review A 106 5, 052410 (2022).
 Tein van der Lugt, Jonathan Barrett, and Giulio Chiribella, "Device-independent certification of indefinite causal order in the quantum switch", Nature Communications 14 1, 5811 (2023).
 Laura J. Henderson, Alessio Belenchia, Esteban Castro-Ruiz, Costantino Budroni, Magdalena Zych, Časlav Brukner, and Robert B. Mann, "Quantum Temporal Superposition: The Case of Quantum Field Theory", Physical Review Letters 125 13, 131602 (2020).
 Pedro R. Dieguez, Vinicius F. Lisboa, and Roberto M. Serra, "Thermal devices powered by generalized measurements with indefinite causal order", Physical Review A 107 1, 012423 (2023).
 Carlos Sabín, "Causality in a Qubit-Based Implementation of a Quantum Switch", Universe 8 5, 269 (2022).
 Robin Lorenz, "Quantum causal models: the merits of the spirit of Reichenbach’s principle for understanding quantum causal structure", Synthese 200 5, 424 (2022).
 Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, and Cyril Branciard, "Communication through coherent control of quantum channels", Quantum 4, 333 (2020).
 Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Cyclic quantum causal models", Nature Communications 12 1, 885 (2021).
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