Licensed Shared Access (LSA) is a new concept proposed by the radio spectrum policy group in order to optimize spectrum usage: a Mobile Network Operator (MNO) can access temporarily to other incumbent's spectrum after obtaining a license. The licensing process is made via an auction mechanism. The mechanisms proposed in the literature for the LSA context are oneshot auction mechanisms which allocate all the available spectrum as a unique block. In this paper, we first show how to increase the performance of those auctions (in terms of revenue, efficiency and fairness of the allocation) while preserving truthful bidding, by splitting spectrum and converting single block auctions into multi-block auctions. Simulation results illustrate how appropriately choosing the number of blocks allows to increase the aforementioned metrics. Second, we show how to convert one-shot mechanisms to equivalent ascending mechanisms (in terms of allocations and payments) so that we add transparency and privacy to the auction.