  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, in the case of a
single owner, this problem can be solved without consensus by using byzantine
quorum systems (requiring a quorum of $2f+1$ validations per transaction).
Nonetheless, the process of validating transactions remains sequential. For
example, if one has a balance of ten coins and intends to make separate
payments of two coins each to two distinct recipients, both transactions must
undergo processing by a common correct validator. On the other hand, these two
transactions are non-conflicting as they do not lead to double spending,
allowing in principle for parallel validation. In this paper, we show that it
is possible to validate payment transactions in parallel with less than $f$
validations per transaction in an asynchronous system, provided that each
transaction spends only a small fraction of a balance. Our solution relies on a
novel class of probabilistic quorum systems that we introduce in this paper,
termed \textit{$(k_1,k_2)$-quorum systems}. In the absence of an adaptive
adversary, \textit{$(k_1,k_2)$-quorum systems} can be used to enable concurrent
and asynchronous validation of up to $k_1$ transactions while preventing
validation of more than $k_2$ transactions. Employing a $(k_1, k_2)$-quorum
system, we introduce protocols enabling a payer to validate multiple
\textit{fractional spending} transactions in parallel with less than $f+1$
validations per transaction. Subsequently, the payer reclaims any remaining
funds through a fully validated transaction, referred to as a
\textit{settlement} transaction.
