Temporal graphs are used to abstractly model real-life networks that are inherently dynamic in nature. Given a static underlying graph $G=(V,E)$, a temporal graph on $G$ is a sequence of snapshots $G_t$, one for each time step $t\geq 1$. In this paper we study stochastic temporal graphs, i.e. stochastic processes $\mathcal{G}$ whose random variables are the snapshots of a temporal graph on $G$. A natural feature observed in various real-life scenarios is a memory effect in the appearance probabilities of particular edges; i.e. the probability an edge $e\in E$ appears at time step $t$ depends on its appearance (or absence) at the previous $k$ steps. In this paper we study the hierarchy of models memory-$k$, addressing this memory effect in an edge-centric network evolution: every edge of $G$ has its own independent probability distribution for its appearance over time. Clearly, for every $k\geq 1$, memory-$(k-1)$ is a special case of memory-$k$. We make a clear distinction between the values $k=0$ ("no memory") and $k\geq 1$ ("some memory"), as in some cases these models exhibit a fundamentally different computational behavior, as our results indicate. For every $k\geq 0$ we investigate the complexity of two naturally related, but fundamentally different, temporal path (journey) problems: MINIMUM ARRIVAL and BEST POLICY. In the first problem we are looking for the expected arrival time of a foremost journey between two designated vertices $s,y$. In the second one we are looking for the arrival time of the best policy for actually choosing a particular $s$-$y$ journey. We present a detailed investigation of the computational landscape of both problems for the different values of memory $k$. Among other results we prove that, surprisingly, MINIMUM ARRIVAL is strictly harder than BEST POLICY; in fact, for $k=0$, MINIMUM ARRIVAL is #P-hard while BEST POLICY is solvable in $O(n^2)$ time.