## 7 Discussion and Future Work  
We provided algorithms in the quantum CONGEST-CLIQUE model for computing approximately optimal Steiner trees and exact directed minimum spanning trees that use asymptotically fewer rounds than their classical known counterparts. As Steiner trees and minimum spanning trees cannot benefit from quantum communication in the CONGEST (non-clique) model, the algorithms reveal how quantum communication can be exploited thanks to the CONGEST-CLIQUE setting. A few open questions remain as well. In particular, there exist many generalizations of the Steiner tree problem, so these may be a natural starting point to attempt to generalize the results. A helpful overview of Steiner-type problems can be found in [20]. Regarding the DMST, it may be difficult to generalize a similar approach to closely related problems. Since the standard MST can be solved in a (relatively small) constant number of rounds in the classical CONGEST-CLIQUE, no significant quantum speedup is possible. Other interesting MST-type problems are the bounded-degree and minimum-degree spanning tree problems. However, even the bounded-degree decision problem on an unweighted graph, \"does \(G\) have a spanning tree of degree at most \(k\)?\" is NP-complete, unlike the DMST, so we suspect that other techniques would need to be employed. Reference [21] provides a classical distributed approximation algorithm for the problem. Additionally, we traced many constants and log factors throughout our description of the above algorithms, which, as shown, would need to be significantly improved for these and related algorithms to be practical. Hence, a natural avenue for future work is to work towards such practical improvements. Beyond the scope of the particular algorithms involved, we hope to help the community recognize the severity in which the practicality of algorithms is affected by logarithmic factors that may be obscured by the \(\tilde{\mathcal{O}}\) notation and, thus, encourage fellow researchers to present the full complexity of their algorithms beyond asymptotics. Particularly in a model such as CONGEST-CLIQUE, where problems can always be solved trivially in \(n\) rounds, these logarithmic factors should clearly not be taken lightly. Further, a question of potential practical interest would be to ask the following: What algorithms solving the discussed problems are the most efficient with respect to rounds needed in the CONGEST-CLIQUE in the regimes of \(n\), in which the discussed algorithms are impractical?  
Conceptualization, E.G.R.; Methodology, P.K. and D.E.B.N.; Software, P.K.; Formal analysis, P.K., D.E.B.N. and Z.G.I.; Investigation, P.K., D.E.B.N., Z.G.I. and E.G.R.; Writing--original draft, P.K.; Writing--review & editing, D.E.B.N., Z.G.I. and E.G.R.; Supervision, D.E.B.N. and E.G.R.; Funding acquisition, E.G.R. All authors have read and agreed to the published version of the manuscript.  
We are grateful for support from the NASA Ames Research Center, from the NASA ScaN program, and from DARPA under IAA 8839, Annex 130. PK and DB acknowledge support from the NASA Academic Mission Services (contract NNA16 BD14C).  
The authors thank [PERSON] for helpful input and discussions regarding the arborescence problem, [PERSON] for ongoing useful discussions, and [PERSON] for helpful feedback on the work.  
The authors declare no conflict of interest.