  Guided by the intuition of coherent superposition of causal relations, recent
works presented quantum processes without classical common-cause and
direct-cause explanation, that is, processes which cannot be written as
probabilistic mixtures of quantum common-cause and quantum direct-cause
relations (CCDC). In this work, we analyze the minimum requirements for a
quantum process to fail to admit a CCDC explanation and present "simple"
processes, which we prove to be the most robust ones against general noise.
These simple processes can be realized by preparing a maximally entangled state
and applying the identity quantum channel, thus not requiring an explicit
coherent mixture of common-cause and direct-cause, exploiting the possibility
of a process to have both relations simultaneously. We then prove that,
although all bipartite direct-cause processes are bipartite separable
operators, there exist bipartite separable processes which are not
direct-cause. This shows that the problem of deciding weather a process is
direct-cause process is not equivalent to entanglement certification and points
out the limitations of entanglement methods to detect non-classical CCDC
processes. We also present a semi-definite programming hierarchy that can
detect and quantify the non-classical CCDC robustnesses of every non-classical
CCDC process. Among other results, our numerical methods allow us to show that
the simple processes presented here are likely to be also the maximally robust
against white noise. Finally, we explore the equivalence between bipartite
direct-cause processes and bipartite processes without quantum memory, to
present a separable process which cannot be realized as a process without
quantum memory.
