  The echo state property (ESP) represents a fundamental concept in the
reservoir computing (RC) framework that ensures output-only training of
reservoir networks by being agnostic to the initial states and far past inputs.
However, the traditional definition of ESP does not describe possible
non-stationary systems in which statistical properties evolve. To address this
issue, we introduce two new categories of ESP: $\textit{non-stationary ESP}$,
designed for potentially non-stationary systems, and $\textit{subspace/subset
ESP}$, designed for systems whose subsystems have ESP. Following the
definitions, we numerically demonstrate the correspondence between
non-stationary ESP in the quantum reservoir computer (QRC) framework with
typical Hamiltonian dynamics and input encoding methods using non-linear
autoregressive moving-average (NARMA) tasks. We also confirm the correspondence
by computing linear/non-linear memory capacities that quantify input-dependent
components within reservoir states. Our study presents a new understanding of
the practical design of QRC and other possibly non-stationary RC systems in
which non-stationary systems and subsystems are exploited.
