The $Lambda$CDM cosmological scenario predicts that our Galaxy should contain hundreds of stellar streams at the solar vicinity, fossil relics of the merging history of the Milky Way and more generally of the hierarchical growth of galaxies. Because of the mixing time scales in the inner Galaxy, it has been claimed that these streams should be difficult to detect in configuration space but can still be identifiable in kinematic-related spaces like the energy/angular momenta spaces, E-Lz and Lperp-Lz, or spaces of orbital/velocity parameters. By means of high-resolution, dissipationless N-body simulations, containing between 25$times10^6$ and 35$times10^6$ particles, we model the accretion of a series of up to four 1:10 mass ratio satellites then up to eight 1:100 satellites and we search systematically for the signature of these accretions in these spaces. In all spaces considered (1) each satellite gives origin to several independent overdensities; (2) overdensities of multiple satellites overlap; (3) satellites of different masses can produce similar substructures; (4) the overlap between the in-situ and the accreted population is considerable everywhere; (5) in-situ stars also form substructures in response to the satellite(s) accretion. These points are valid even if the search is restricted to kinematically-selected halo stars only. As we are now entering the Gaia era, our results warn that an extreme caution must be employed before interpreting overdensities in any of those spaces as evidence of relics of accreted satellites. Reconstructing the accretion history of our Galaxy will require a substantial amount of accurate spectroscopic data, that, complemented by the kinematic information, will possibly allow us to (chemically) identify accreted streams and measure their orbital properties. (abridged)
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