  In a Bell experiment, it is natural to seek a causal account of correlations
wherein only a common cause acts on the outcomes. For this causal structure,
Bell inequality violations can be explained only if causal dependencies are
modelled as intrinsically quantum. There also exists a vast landscape of causal
structures beyond Bell that can witness nonclassicality, in some cases without
even requiring free external inputs. Here, we undertake a photonic experiment
realizing one such example: the triangle causal network, consisting of three
measurement stations pairwise connected by common causes and no external
inputs. To demonstrate the nonclassicality of the data, we adapt and improve
three known techniques: (i) a machine-learning-based heuristic test, (ii) a
data-seeded inflation technique generating polynomial Bell-type inequalities
and (iii) entropic inequalities. The demonstrated experimental and data
analysis tools are broadly applicable paving the way for future networks of
growing complexity.
