Quantum systems displaying many accessible levels could be very powerful units of forthcoming quantum computing architectures. Indeed, the large number of available states could significantly simplify the actual implementation of several algorithms. Here we show that artificial molecular spins are particularly suitable to realize such a platform. In particular, multispin molecules with competing interactions provide a large number of low-energy multiplets in which decoherence is strongly suppressed compared to a single spin S and does not increase with the system size. This feature, combined with the proper connectivity between the multiplets, enables the implementation of complex operations with remarkable fidelity, thus fully unleashing the potential of the molecular approach. We demonstrate the power of this approach by numerically simulating the implementation of one- and two-qudit gates on realistic molecular systems.