  In this work, we show the importance of searches for heavy resonant scalars
($H$) and pseudoscalars ($A$). Taking cue from the present searches, we make
projections for searches in an extended scalar sector at the high luminosity
run of the Large Hadron Collider. We study the three most relevant search
channels, \textit{i.e.}, $H \to hh, \; H/A \to t\bar{t}$ and $b\bar{b}H/A$.
Upon studying multifarious final states for the resonant double Higgs
production, we find that the $b\bar{b}\gamma\gamma$ ($\sigma(pp \to H \to hh)
\in [81.27,14.45]$ fb for $m_H \in [300,600]$ GeV at 95\% C.L.) and
$b\bar{b}b\bar{b}$ ($[5.4,2.5]$ fb for $m_H \in [800,1000]$ GeV at 95\% C.L.)
channels are the most constraining. For the $b\bar{b}H$ channel, we can exclude
$\sigma(pp \to b\bar{b}H) \in [22.2,3.7]$ fb for $m_H \in [300,500]$ GeV.
Finally, we consider the phenomenological Minimal Supersymmetric Standard Model
as an example and impose various present constraints and our future direct
search-limits and obtain strong constraints on the $m_A-\tan{\beta}$ parameter
space, where $m_A$ and $\tan{\beta}$ are respectively the mass of the
pseudoscalar and the ratio of the vacuum expectation values of the two Higgs
doublets. Assuming that the heavy Higgs boson decays only to Standard Model
(SM) states, we find that the $H \to hh \to b\bar{b}\gamma\gamma$ ($H \to
t\bar{t}$) channel excludes $\tan{\beta}$ as low as 4 ($m_A \in [400,800]$ GeV)
at 95\% CL. This weakens up to $\sim 5.5$ when the $b\bar{b}H$ channel
dominates. Upon allowing for non-SM decay modes, the limits weaken.
