  The combination of interactions and static gauge fields plays a pivotal role
in our understanding of strongly-correlated quantum matter. Cold atomic gases
endowed with a synthetic dimension are emerging as an ideal platform to
experimentally address this interplay in quasi-one-dimensional systems. A
fundamental question is whether these setups can give access to pristine
two-dimensional phenomena, such as the fractional quantum Hall effect, and how.
We show that unambiguous signatures of bosonic and fermionic Laughlin-like
states can be observed and characterized in synthetic ladders. We theoretically
diagnose these Laughlin-like states focusing on the chiral current flowing in
the ladder, on the central charge of the low-energy theory, and on the
properties of the entanglement entropy. Remarkably, Laughlin-like states are
separated from conventional liquids by Lifschitz-type transitions,
characterized by sharp discontinuities in the current profiles, which we
address using extensive simulations based on matrix-product states. Our work
provides a qualitative and quantitative guideline towards the observability and
understanding of strongly-correlated states of matter in synthetic ladders. In
particular, we unveil how state-of-the-art experimental settings constitute an
ideal starting point to progressively tackle two-dimensional strongly
interacting systems from a ladder viewpoint, opening a new perspective for the
observation of non-Abelian states of matter.
