  We consider the problem of supporting payment transactions in an asynchronous
system in which up to $f$ validators are subject to Byzantine failures under
the control of an adaptive adversary. It was shown that this problem can be
solved without consensus by using byzantine quorum systems (requiring at least
$2f+1$ validations per transaction in asynchronous systems). We show that it is
possible to validate transactions in parallel with less than $f$ validations
per transaction if each transaction spends no more that a small fraction of a
balance. Our solution relies on a novel quorum system that we introduce in this
paper and that we call $(k_1,k_2)$-quorum systems. In the presence of a
non-adaptive adversary, these systems can be used to allow up to $k_1$
transactions to be validated concurrently and asynchronously but prevent more
than $k_2$ transactions from being validated. If the adversary is adaptive,
these systems can be used to allow $k_1$ transaction to be validated and
prevent more than $k'_2 > k_2$ transactions from being validated, the
difference $k'_2-k_2$ being dependent on the quorum system's {\em validation
slack}, which we define in this paper. Using $(k_1,k_2)$-quorum systems, a
payer can execute multiple partial spending transactions to spend a portion of
its initial balance with less than full quorum validation (less than $f$
validations per transaction) then reclaim any remaining funds using one fully
validated transaction, which we call a {\em settlement} transaction.
