  $\newcommand{\F}{\mathbb{F}}$We study the Boolean function parameters
sensitivity ($s$), block sensitivity ($bs$), and alternation ($alt$) under
specially designed affine transforms. For a function $f:\F_2^n\to \{0,1\}$, and
$A=Mx+b$ for $M \in \F_2^{n\times n}$ and $b\in \F_2^n$, the result of the
transformation $g$ is defined as $\forall x\in\F_2^n, g(x)=f(Mx+b)$.
  We study alternation under linear shifts ($M$ is the identity matrix) called
the shift invariant alternation (denoted by $salt(f)$). We exhibit an explicit
family of functions for which $salt(f)$ is $2^{\Omega(s(f))}$. We show an
affine transform $A$, such that the corresponding function $g$ satisfies
$bs(f,0^n) \le s(g)$, using which we proving that for $F(x,y)=f(x\land y)$, the
bounded error quantum communication complexity of $F$ with prior entanglement,
$Q^*_{1/3}(F)=\Omega(\sqrt{bs(f,0^n)})$. Our proof builds on ideas from
Sherstov (2010) where we use specific properties of the above affine
transformation. We show,
  * For a prime $p$ and $0<\epsilon<1$, any $f$ with
$deg_p(f)\le(1-\epsilon)\log n$ must satisfy $Q^*_{1/3}(F) =
\Omega(\frac{n^{\epsilon/2}}{\log n})$. Here, $deg_p(f)$ denotes the degree of
the multilinear polynomial of $f$ over $\F_p$.
  * For any $f$ such that there exists primes $p$ and $q$ with $deg_q(f) \ge
\Omega(deg_p(f)^\delta)$ for $\delta > 2$, the deterministic communication
complexity - $D(F)$ and $Q^*_{1/3}(F)$ are polynomially related. In particular,
this holds when $deg_p(f) = O(1)$. Thus, for this class of functions, this
answers an open question (see Buhrman and deWolf (2001)) about the relation
between the two measures.
  We construct linear transformation $A$, such that $g$ satisfies, $alt(f) \le
2s(g)+1$. Using this, we exhibit a family of Boolean functions that rule out a
potential approach to settle the XOR Log-Rank conjecture via a proof of
Sensitivity conjecture [Hao Huang (2019)].
