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The Medium.com article Mars Phoenix Lander, 10 Years Later shows several remarkable images and discoveries on Mars by the Mars Phoenix Lander circa 2008.

One image (shown below) shows what looks like droplets of liquid water, condensed on the surface of one of the lander's legs.

The article says (emphasis mine):

Shortly after landing, the camera on Phoenix’s robotic arm captured views of blobs of material on one of the landing struts. Over time, these blobs moved, darkened, and coalesced, behaving like droplets of liquid water. The hypothesis here was that these blobs “splashed up” on the struts when the descent thrusters melted the ice exposed upon landing mentioned above.

But if liquid water isn’t stable on the martian surface, how did Phoenix observe liquid water on Mars? The key here lies in salt. If you live anywhere that gets snow, you’re probably familiar with salt as a de-icer for roads, sidewalks, etc. Salt lowers the freezing point of water, allowing it to remain liquid at temperatures lower than that of non-salty water. For example, pure water freezes at 0 °C/32 °F, but ocean saltwater freezes around −2 °C/28.4 °F. While the de-icing salts you get at the hardware store lower the freezing point by a few degrees, more exotic salts can lower the freezing point as much as −70 °C/−89 °F! Phoenix discovered some of these exotic salts in the soil around the lander—in particular, magnesium perchlorate. (note, minor editorial changes have been made)

Question: Which "exotic salt" can lower water's freezing point by −70 °C?

Is it in fact magnesium perchlorate (which was found on Mars) or is it a different salt?

Blobs of possible brine (really salty water) imaged on one of Phoenix’s landing struts shortly after arriving on Mars. Credit: NASA/JPL-Caltech/University of Arizona/Max Planck Institute

I recently got a chance to attend a talk by someone who was working on developing analytical instrumentation on Mars. The interesting story is that the initial results by ion-selective electrode was that Mars soil is full of nitrates. Nobody knew on Earth that the nitrate ion selective electrode is far more responsive to perchlorate than nitrate. After learning this, it was an eye opener for analytical chemists! Now they wish to use chromatography rather than electrochemistry. So this was a good lesson for us on Earth.

Now that they know it is a perchlorate ion, people did some studies on supercooled brines. See this paper: The formation of supercooled brines, viscous liquids, and low-temperature perchlorate glasses in aqueous solutions relevant to Mars J.D. Toner, D.C. Catling, and B. Light, Icarus, 233, 1 May 2014, pp 36-47 (also available here). They clearly show that if calcium or magnesium perchlorates are slowly cooled, one can get supercooled brines up to -120 Celcius. This is a rather amazing finding. They call it a glassy state.

Your Question: Which "exotic salt" can lower water's freezing point by $pu-70 ^circ C$?

Here is your "exotic compound" although it is not a salt by definition. It is a base: Aqua ammonia, also called ammoniacal liquor, ammonia liquor, or ammonia water, is produced by dissolving ammonia gas ($ceNH3$) in water. The proper chemical name of aqua ammonia is ammonium hydroxide ($ceNH4OH$), which is in following equilibrium with water:
$$ceNH3 + H2O <=> NH4+ + OH-$$

Ammonia is very soluble in water: According to Wikipedia, its solubility in water is $47% (w/w)$ at $pu0 ^circ C$, $31% (w/w)$ at $pu25 ^circ C$, and $18% (w/w)$ at $pu50 ^circ C$. Therefore it is ideal to cause large freezing point depression since its solubility increases with decreasing temperature. Now, let's see how are the freezing points of aqua ammonia solutions behave with increasing concentrations. The large scale manufacturer of aqua ammonia, Tanner Industries, listed following values of boiling and freezing points of various solutions in its Customer Manual:
$$
beginarrayccc \hline
% ceNH3 text (by weight) & textApprox. Boiling point & text Approx. Freezing point \hline
23.52 & pu103 ^circ F: (pu39.4 ^circ C) & pu-56 ^circ F: ( pu-48.9 ^circ C)\
25.48 & pu95 ^circ F: (pu35.0 ^circ C) & pu-69 ^circ F : ( pu-56.1 ^circ C)\
27.44 & pu88 ^circ F : (pu31.1 ^circ C) & pu-89 ^circ F: ( pu-67.2 ^circ C)\
29.40 & pu85 ^circ F: (pu29.4 ^circ C) & pu-110 ^circ F : ( pu-78.9 ^circ C)\
31.36 & pu73 ^circ F: (pu22.8 ^circ C) & pu-123 ^circ F : ( pu-86.1 ^circ C)\
33.32 & pu66 ^circ F : (pu18.9 ^circ C) & pu-148 ^circ F: ( pu-100 ^circ C)\hline
endarray
$$

Accordingly, anything between $29-33%$ of aqua ammonia solution would do the job.