The performance of earlier protocols was connected in a similar way to the monotone formula complexity of these functions. We show that the well known separation between monotone span programs and monotone formulae carries over to the types of functions occurring here, which implies that monotone span programs give rise to strictly more efficient verifiable secret sharing protocols that what was known earlier.
If the separation we show could be proved to hold, also if the span program is required to have a special "multiplication property", this would imply that span programs can make general multiparty computations more efficient. This is open at the time of writing.