  Decentralized payment systems such as Bitcoin have become massively popular
in the last few years, yet there is still much to be done in understanding
their formal properties. The vast majority of decentralized payment systems
work by achieving consensus on the state of the network; a natural question to
therefore ask is whether this consensus is necessary. In this paper, we
formally define a model of payment systems, and present two main results. In
Theorem 1, we show that even though there exists a single step black box
reduction from Payment Systems to Byzantine Broadcast, there does not exist any
black box reduction in the other direction which is significantly better than a
trivial reduction. In Theorem 2, we show how to construct Payment Systems which
only require a very small number of messages to be sent per transaction. In
particular, global consensus about which transactions have occurred is not
necessary for payments in this model. We then show a relation between the
construction in Theorem 2 and the Lightning Network, relating the formal model
constructions we have given to a practical algorithm proposed by the
cryptocurrency community.
