  We consider fractional Hartree and cubic nonlinear Schr\"odinger equations on
Euclidean space $\mathbb R^d$ and on torus $\mathbb T^d$. We establish norm
inflation (a stronger phenomena than standard ill-posedness) at every initial
data in Fourier amalgam spaces with negative regularity. In particular, these
spaces include Fourier-Lebesgue, modulation and Sobolev spaces. We further show
that this can be even worse by exhibiting norm inflation with an infinite loss
of regularity. To establish these phenomena, we employ a Fourier analytic
approach and introduce new resonant sets corresponding to the fractional
dispersion $(-\Delta)^{\alpha/2}$. In particular, when dispersion index
$\alpha$ is large enough, we obtain norm inflation {above} scaling critical
regularity in some of these spaces. It turns out that our approach could treat
both equations (Hartree and power-type NLS) in a unified manner. The method
should also work for a broader range of nonlinear equations with Hartree-type
nonlinearity.
