It is shown that dipolar and weak superexchange interactions between the spin systems of singlemolecule magnets (SMM) play an important role in the relaxation of magnetization. These interactions can reduce or increase resonant tunneling. At certain external fields, the mechanism of spin-spin cross-relaxation (SSCR) can lead to quantum resonances which can show up in hysteresis loop measurements as well defined steps. A Mn4 SMM is used as a model system to study the SSCR which plays also an important role for other SMMs like Mn12 or Fe8.PACS numbers: PACS numbers: 75.45.+j, 75.60.Ej Single-molecule magnets (SMMs) [1,2] are one of the best systems for studying quantum tunneling of large moments [3]. Each molecule functions as a nanoscale, single-domain magnetic particle that, below its blocking temperature, exhibits the classical macroscale property of a magnet, namely magnetization hysteresis. Such a molecule straddles the classical/quantum interface in also displaying quantum tunneling of magnetization [4,5,6,7,8,9,10] and quantum phase interference [11,12,13]. A quantitative understanding of the mechanism of magnetization tunneling is being developed. For example, the width of tunnel transitions are in general larger than expected from dipolar interactions. Crystal defects may play an important role: loss or disorder of solvent molecules, and even dislocations have been proposed [14].Since SMMs occur as assemblies in crystals, there is the possibility of a small electronic interaction of adjacent molecules. This leads to very small superexchange interactions (or exchange interactions, for short) that depend strongly on the distance and the non-magnetic atoms in the exchange pathway. Up to now, such an intermolecular exchange interaction has been assumed to be negligibly small. However, our recent studies on several SMMs suggest that in most SMMs exchange interactions lead to a significant influence on the tunnel process. Recently, this intermolecular exchange interaction was used to couple antiferromagnetically two SMMs, each acting as a bias on its neighbor, resulting in quantum behavior different from that of individual SMMs [15].The main difference between dipole and exchange interactions are: (i) dipole interactions are long range whereas exchange interactions are usually short range; (ii) exchange interactions can be much stronger than dipolar interactions; (iii) whereas the sign of a dipolar interaction can be determined easily, that of exchange depends strongly on electronic details and is very difficult to predict; and (iv) dipolar interactions depend strongly on the spin ground state S, whereas exchange interactions depend strongly on the single-ion spin states. For example, intermolecular dipolar interactions can be neglected for antiferromagnetic SMMs with S = 0, whereas intermolecular exchange interactions can still be important and act as a source of decoherence.In this letter we show that dipolar and/or exchange interactions can lead to collective quantum processes. The one-body tunnel picture...