  Quantum theory admits ensembles of quantum nonlocality without entanglement
(QNLWE). These ensembles consist of seemingly classical states (they are
perfectly distinguishable and non-entangled) that cannot be perfectly
discriminated with local operations and classical communication (LOCC). Here,
we analyze QNLWE from a causal perspective, and show how to perfectly
discriminate some of these ensembles using local operations and classical
communication without definite causal order. Specifically, three parties with
access to an instance of indefinite causal order-the AF/BW process-can
perfectly discriminate the states in a QNLWE ensemble--the SHIFT ensemble--with
local operations. Hence, this type of quantum nonlocality disappears at the
expense of definite causal order while retaining classical communication. Our
results thereby leverage the fact that LOCC is a conjunction of three
constraints: local operations, classical communication, and definite causal
order. Moreover, we show how multipartite generalizations of the AF/BW process
are transformed into multiqubit ensembles that exhibit QNLWE. Such ensembles
are of independent interest for cryptographic protocols and for the study of
separable quantum operations unachievable with LOCC.
