  The most promising mechanisms for the formation of Majorana bound states
(MBSs) in condensed matter systems involve one-dimensional systems (such as
semiconductor nanowires, magnetic chains, and quantum spin Hall insulator
(QSHI) edges) proximitized to superconducting materials. The choice between
each of these options involves trade-offs between several factors such as
reproducibility of results, system tunability, and robustness of the resulting
MBS. In this article, we propose that a combination of two of these systems,
namely a magnetic chain deposited on a QSHI edge in contact with a
superconducting surface, offers a better choice of tunability and MBS
robustness compared to magnetic chain deposited on bulk. We study how the QSHI
edge interacts with the magnetic chain, and see how the topological phase is
affected by edge proximity. We show that MBSs near the edge can be realized
with lower chemical potential and Zeeman field than the ones inside the bulk,
independently of the chain's magnetic order (ferromagnetic or spiral order).
Different magnetic orderings in the chain modify the overall phase diagram,
even suppressing the boundless topological phase found in the bulk for chains
located at the QSHI edge. Moreover, we quantify the "quality" of MBSs by
calculating the Majorana Polarization (MP) for different configurations. For
chains located at the edge, the MP is close to its maximum value already for
short chains. For chains located away from the edge, longer chains are needed
to attain the same quality as chains located at the edge. The MP also
oscillates in phase with the in-gap states, which is relatively unexpected as
peaks in the energy spectrum corresponds to stronger overlap of MBSs.
