  We prove general equidistribution statements (both conditional and
unconditional) relating to the Fourier coefficients of arithmetically
normalized holomorphic Hecke cusp forms $f_1,\ldots,f_k$ without complex
multiplication, of equal weight, (possibly different) squarefree level and
trivial nebentypus. As a first application, we show that for the Ramanujan
$\tau$ function and any admissible $k$-tuple of distinct non-negative integers
$a_1,\ldots,a_k$ the set $$ \{n \in \mathbb{N} : |\tau(n+a_1)| < \cdots <
|\tau(n+a_k)|\} $$ has positive natural density. This result improves upon
recent work of Bilu, Deshouillers, Gun and Luca [Compos. Math. (2018), no. 11,
2441-2461]. Secondly, we make progress towards understanding the signed version
by showing that $$ \{n \in \mathbb{N} : \tau(n+a_1) < \tau(n+a_2) <
\tau(n+a_3)\} $$ has positive relative upper density at least $1/6$ for any
admissible triple of distinct non-negative integers $(a_1,a_2,a_3).$ More
generally, for such chains of inequalities of length $k > 3$ we show that under
the assumption of Elliott's conjecture on correlations of multiplicative
functions, the relative natural density of this set is $1/k!.$ Previously
results of such type were known for $k\le 2$ as consequences of works by Serre
and by Matom\"{a}ki and Radziwill. Our results rely crucially on several key
ingredients: i) a multivariate Erd\H{o}s-Kac type theorem for the function $n
\mapsto \log|\tau(n)|$, conditioned on $n$ belonging to the set of
non-vanishing of $\tau$, generalizing work of Luca, Radziwill and Shparlinski;
ii) the recent breakthrough of Newton and Thorne on the functoriality of
symmetric power $L$-functions for $\text{GL}(n)$ for all $n \geq 2$ and its
application to quantitative forms of the Sato-Tate conjecture; and iii) the
work of Tao and Ter\"{a}v\"{a}inen on the logarithmic Elliott conjecture.
