Table 1.

Summary of the main differences between the microscopic and collisional approaches. The stability condition is very similar in the two approaches, and the main difference is given only by the discrete nature of the collision model. The Born and Markov approximations are instead where the differences between the two approaches become the most evident: in facts, while in the microscopic approach one needs to enforce both during the derivation of the master equation, in the collisional approach these two approximations are already encompassed in the initial conditions, namely the fact that all the ancillas do not share any initial correlation and that they do not interact with each other during the dynamics. As a matter of fact, non-Markovianity is usually introduced in collision models by allowing interactions between the ancillas so that some memory effect is retained. Finally, also the secular approximation is identical in both approaches, as it deals with the ratio between the typical timescale of the system S and the one of the joint system. However, as highlighted in the main text, the secular approximation is usually not needed in most collision models, as the interaction Hamiltonian is chosen so as to conserve the particle number.

Approximation	Microscopic Approach	Collisional Approach
Stability condition	${\mathrm{Tr}}_{\mathcal{E}}\{{\widehat{H}}_{S\mathcal{E}}{\widehat{\rho }}_{S\mathcal{E}}\left(0\right)\}=0$	${\mathrm{Tr}}_{E_i}\{{\widehat{H}}_{S{E}_i}({\widehat{\rho }}_S\left((i-1)\delta t\right)\otimes {\widehat{\eta }}_{E_i})\}=0$
Born approximation	${\widehat{\rho }}_{S\mathcal{E}}\left(t\right)\simeq {\widehat{\rho }}_S\left(t\right)\otimes {\widehat{\eta }}_{\mathcal{E}}$	Not needed, as it is already encompassed by the assumption of tensorized ancillas
Markov approximation	${\widehat{\rho }}_S\left(s\right)\to {\widehat{\rho }}_S\left(t\right)$	Not needed, as the ancillas do not interact with each other
Secular approximation	${\tau }_S\simeq {|\omega -{\omega }^{\prime }|}^{-1}<<{\tau }_R$	Not needed, as each collision is already completely positive