- In many interesting models, including superstring theories, a negative vacuum en- ergy is predicted. Although this effect is usually regarded as undesirable from a cos- mological point of view, we show that this can be the basis for a new approach to the cosmology of the early Universe. In the framework of quantum cosmology (in higher dimensions) when we consider a negative cosmological constant and matter that could be dust or, alternatively, coherent excitations of a scalar field, the role of cosmic time can be understood. Then we can predict the existence of a "quantum inflationary phase" for some dimensions and a simultaneous "quantum deflationary phase" for the remaining dimensions. We discuss how it may be possible to exit from this inflation- compactification era to a phase with zero cosmological constant which allows a classical description at late times.