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 \opening{To the editors of Physical Review,}
 
 The subject of `complex landscapes,' which started in the spin-glass
-literature, is concerned with functions (landscapes) of many variables, having
-a multiplicity of minima, which are the objects of interest. Apart from its
-obvious interest for glassy systems, it has found a myriad applications in many
-domains: computer science, ecology, economics, biology.
-\footfullcite{Mezard_2009_Information}
+literature, is concerned with functions (landscapes) of many variables having
+a multiplicity of minima. Apart from its obvious relevance to glassy systems,
+it has found applications in many domains: computer science, ecology,
+economics, and biology, to name a few.  \footfullcite{Mezard_2009_Information}
 
-In the last few years, a renewed interest has developed for landscapes for
-which the variables are complex. There are a few reasons for this: {\em i)} in
-computational physics, there is the main obstacle of the `sign problem', and a
-strategy has emerged to attack it deforming the sampling space into complex
-variables. This is a most natural and promising path, and any progress made
-will have game-changing impact in solid state physics and lattice-QCD.
-\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The}  {\em ii)} At a more basic
-level, following the seminal work of E.~Witten,
-\footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a flurry of activity
-concerning the very definition of quantum mechanics, which requires also that
-one move into the complex plane. 
+In the last few years, a renewed interest has developed in landscapes for
+which the variables are complex. There are a couple reasons for this: in
+computational physics the `sign problem' is a major obstacle, and a strategy
+has emerged to attack it by deforming the sampling space into complex variables.
+This is a most natural and promising path, and any progress made will have
+game-changing impact in solid state physics and lattice QCD.
+\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} At a more basic
+level, following the seminal work of E.~Witten, \footfullcite{Witten_2010_A,
+Witten_2011_Analytic} there has been a flurry of activity concerning the very
+definition of quantum mechanics, which also requires that one move into the
+complex plane. 
 
-In all these cases, just like in the real case, one needs to know the structure
-of the `landscape.' where are the saddle points and how they are connected,
-typical questions of `complexity.'  However, to the best of our knowledge,
+In these cases, just as in the real case, one needs to understand the structure
+of the `landscape,' like the location of saddle points, how they are connected,
+and typical questions of `complexity.'  However, to the best of our knowledge,
 there are no studies extending the methods of the theory of complexity to
 complex variables.  We believe our paper will open a field that may find
 numerous applications and will widen our theoretical view of complexity in