  It has been recently shown that under pressure trilayer Ruddlesden-Popper
nickelate La$_4$Ni$_3$O$_{10}$ (LNO) becomes superconducting below a critical
temperature $\sim$20 K, in addition to the infinite-layer and bilayer systems.
Motivated by this observation, we explore the effects of electron correlations
on its electronic structure and magnetic properties using the advanced
DFT+dynamical mean-field theory approach. Our results for the normal-state
electronic structure and correlation effects in LNO have much in common with
the infinite-layer and bilayer nickelates, with a remarkable site- and
orbital-dependent renormalizations of the Ni $3d$ bands and notable incoherence
of the Ni $d_{3z^2-r^2}$ states, caused by correlation effects. Our analysis of
the Fermi surface and magnetic correlations suggests the emergence of competing
spin and charge stripe states, implying the importance of in-plane spin
fluctuations to explain superconductivity in this material.
